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\title[Galois Module Construction and Classification]
{Construction and Classification of Some Galois Modules}

\author[J\'{a}n Min\'{a}\v{c}]{J\'an Min\'a\v{c}$^{*\dagger}$}
\address{Department of Mathematics, Middlesex College,
\ University of Western Ontario, London, Ontario \ N6A 5B7 \
CANADA}
\thanks{$^*$Research supported in part by the Natural Sciences and
Engineering Research Council of Canada, and by the special Dean of
Science Fund at the University of Western Ontario.}
\thanks{$^\dag$Supported by the Mathematical Sciences Research
Institute, Berkeley}
\email{minac@uwo.ca}

\author[John Swallow]{John Swallow$^\ddag$}
\address{Department of Mathematics, Davidson College, Box 7046,
Davidson, North Carolina \ 28035-7046 \ USA}
\thanks{$^\ddag$Research supported in part by National Security
Agency grant MDA904-02-1-0061.} \email{joswallow@davidson.edu}

\begin{abstract}
In our previous paper we describe the Galois module structures of
$p$th-power class groups $K^\times/{K^{\times p}}$, where $K/F$ is
a cyclic extension of degree $p$ over a field $F$ containing a
primitive $p$th root of unity. Our description relies upon
arithmetic invariants associated with $K/F$. Here we construct
field extensions $K/F$ with prescribed arithmetic invariants, thus
completing our classification of Galois modules
$K^{\times}/K^{\times p}$.

\end{abstract}

\date\today

\maketitle

Let $F$ be a field of characteristic not $p$ containing a
primitive $p$th root of unity $\xi_p$.  For a cyclic field
extension $K/F$ with Galois group $\Gal(K/F)$ of order $p$, let $J
= {K^\times}/{K^{\times p}}$, and let $N$ denote the norm map from
$K$ to $F$.

In \cite{MS}, we proved that the structure of the
$\Fp[\Gal(K/F)]$-module $J$ is determined by the
following three arithmetic invariants:

\begin{itemize}
    \item $d=d(K/F):=\dim_{\Fp} {F^\times}/{N(K^\times)}$,
    \item $e=e(K/F):=\dim_{\Fp} N(K^{\times})/F^{\times p}$, and
    \item $\Upsilon(K/F):=$ $1$ or $0$ according to whether
    $\xi_p\in N(K^\times)$ or not.
\end{itemize}

Now if $G=\Z/p\Z$, then $J$ may be considered an $\Fp[G]$-module
via any isomorphism $G\cong \Gal(K/F)$, and the module structure
of $J$ is independent of the choice of isomorphism. It is a
fundamental problem to classify the isomorphism classes of modules
$J$ for all $K/F$ in our context. This problem is solved in
Theorem~\ref{th:mainth} below. Corollaries~1 and 2 of
Theorem~\ref{th:mainth} describe all modules $J$ in an explicit
way.

In the following theorem we determine the sets of invariants $(d,
e, \Upsilon)$ which may be realized by an extension $K/F$ and in
so doing classify all $\Fp[G]$-modules $J=J(K/F)$ up to
isomorphism.

\begin{theorem}\label{th:mainth}
    Let $p$ be a prime number. For arbitrary cardinal numbers $d$,
    $e$, and for $\Upsilon\in \{0,1\}$, there exists a cyclic
    field extension $K/F$ of degree $p$ containing a primitive
    $p$th root of unity with invariants $(d, e, \Upsilon)$ if and
    only if
    \begin{itemize}
        \item if $\Upsilon=0$, then $1\le d$,
        \item if $p>2$ then $1\leq e$, and
    \item if $p=2$ and $\Upsilon=1$ then $1\leq e$.
    \end{itemize}
\end{theorem}

From the theorem above and from \cite[Theorem~3 and
Corollary~2]{MS} we immediately obtain the following corollaries.
We denote by $M_{i,j}$ the $j$th cyclic module $\Fp[G]$ such that
$\dim_{\Fp}M_{i,j}=i$, where $j$ is a suitable index.

\begin{corollary} \label{co:maincor1}
    Let $p>2$ be a prime number, and let $G$ be a cyclic
    group of order $p$. Then an $\Fp[G]$-module $J$ is realizable
    as an $\Fp[\Gal(K/F)]$-module $K^{\times}/K^{\times p}$ for
    some cyclic $G$-extension $K/F$ such that $F$ contains a
    primitive $p$th root of unity if and only if there exist
    cardinal numbers $d$, $e$, and $\Upsilon\in\{0,1\}$ such that
    \begin{itemize}
        \item[$(i)$] If $\Upsilon=0$, then $1\le d$;
        \item[$(ii)$] $1\leq e$; and
        \item[$(iii)$]
        \begin{equation*}
            J=\left(\bigoplus_{j\in\Kf_1}M_{1,j}\right)
            \bigoplus\left(\bigoplus_{j\in\Kf_2}M_{2,j}\right)
            \bigoplus\left(\bigoplus_{j\in\Kf_p}
            M_{p,j}\right),
        \end{equation*}
        where
        \medskip
        \begin{enumerate}
            \item[$(1)$] $\vert\Kf_1\vert+1=2\Upsilon+d$,
            \item[$(2)$] $\vert\Kf_2\vert=1-\Upsilon$, and
            \item[$(p)$] $\vert\Kf_p\vert+1=e$.
        \end{enumerate}
    \end{itemize}
    The invariants $d$, $e$, and $\Upsilon$ determine the module $J$
    uniquely.
\end{corollary}

For $p=2$ using \cite[Theorem~3 and Corollary~3]{MS}, along with
our theorem above, we obtain the next corollary.

\begin{corollary} \label{co:maincor2}
    Now let $G$ be a cyclic group of order $2$. Then an
    $\F_2[G]$-module $J$ is realizable as an
    $\F_2[\Gal(K/F)]$-module $K^{\times}/K^{\times 2}$ for some
    quadratic extension $K/F$ with its arithmetic invariants
    $d(K/F)$, $e(K/F)$, and $\Upsilon(K/F)$ coinciding with $d$,
    $e$, and $\Upsilon\in\{0,1\}$, respectively, if and only if $d$, $e$, 
    and $\Upsilon$ satisfy the conditions below.

    \begin{itemize}
        \item if $\Upsilon=0$, then $1\leq d$,
    \item if $\Upsilon=1$, then $1\leq e$.
    \end{itemize}

    In this case
    \begin{equation*}
        J=\left(\bigoplus_{j\in\Kf_1}M_{1,j}\right)
        \bigoplus \left(\bigoplus_{j\in\Kf_2}M_{2,j}\right)
    \end{equation*}
    where
    \begin{enumerate}
        \item[(1)] $\vert\Kf_1\vert+1=2\Upsilon+d$ and
        \item[(2)] $\vert\Kf_2\vert+\Upsilon=e$.
    \end{enumerate}

    Moreover, such a module $J$ is determined uniquely by the
    invariants $2\Upsilon+d$ and by $e-\Upsilon$ if $e$ is finite
    and by $e$ alone if $e$ is infinite.
\end{corollary}

If $p>2$ then two $\Fp[G]$-modules are isomorphic if and only if
their invariants $d$, $e$, and $\Upsilon$ are the same, by
\cite[Corollary~2]{MS}. Thus we see in particular that if $p>2$,
then the arithmetic invariants of $J$ depend only upon the
isomorphism type of the $\Fp[G]$-module $J$.

In the case $p=2$, we see from \cite[Corollary~2]{MS} again that
the arithmetic invariants $d$, $e$, and $\Upsilon$ determine our
module $\F_2[G]$, but two isomorphic $\F_2[G]$-modules may have
different arithmetic invariants; see \cite[Corollary~3]{MS}. Here
is a very simple, concrete example illustrating this possibility.
Let $K_1/F_1$ be a quadratic extension of finite fields of
characteristic not $2$, $F_2=\R((t))$ be a field of power series
with coefficients in real numbers $\R$, and $K_2=F_2(\sqrt{-1})$.
Then both modules $K_{1}^{\times}/K_{1}^{\times 2}$ and
$K_{2}^{\times}/K_{2}^{\times 2}$ are isomorphic to a trivial
$\F_2[G]$-module $\F_2$, but their arithmetic invariants $(d_i,
e_i, \Upsilon_i)$ are $(0, 1, 1)$ for $i=1$ and $(2, 0, 0)$ for
$i=2$.

\section{Notation and Strategy} \label{S1}

In all that follows $F$ denotes a field, $F^{\times}=
F\setminus\{0\}$ the multiplicative group of $F$, $p$ a prime
number, and $F^{\times}/F^{\times p}$ the group of $p$th-power
classes of $F$. For each $f\in F^{\times}$ we denote by $[f]$ the
class of $f$ in $F^{\times}/F^{\times p}$. For each subset $A$ of
$F^{\times}$ we denote by $[A]$ the set of classes $\{[a]\ \vert\
a\in A\}$ and by $\mo[A]\mc$ the subgroup of $F^{\times}/F^{\times
p}$ generated by $[A]$.

We denote by $\xi_p$ a primitive $p$th root of unity in $F$. (Some
fields will be assumed to contain such a primitive $p$th root; for
the other fields in this paper, we will prove that a primitive
$p$th root is contained in the field.)  Observe that our
assumption that there exists a primitive $p$th root of unity
implies that $\chr(F)\neq p$.

For a Galois extension $K/F$, $\Gal(K/F)$ denotes the Galois group
and $N_{K/F}$ denotes the norm map from $K$ to $F$.  We denote by
$F^s$ the separable closure of $F$ and $G_F$ the absolute Galois
group $\Gal(F^s/F)$.  As usual, $H^i(G_F, \Fp)$ are Galois
cohomology groups of $F$ with coefficients in $\Fp$.  Since all
absolute Galois groups will be pro-$p$-groups, all considered
$\Fp$ modules are trivial.  Finally, let $\vert B \vert$ be the
cardinal number of a set $B$.

First observe that the conditions on $d$, $e$, and $\Upsilon$
listed in our theorem above are necessary:
\begin{enumerate}
    \item If $\Upsilon=0$, then $\xi_p\notin N_{K/F}(K^{\times})$
    and hence
    \begin{equation*}
        d =\dim_{\Fp} F^{\times}/N_{K/F}(K^{\times}) \geq 1;
    \end{equation*}
    \item If $p>2$ and $K=F(\root{p}\of{a})$ for a suitable $a\in
    F^{\times}\setminus F^{\times p}$, then $a=
    N_{K/F}(\root{p}\of{a})$ and hence
    \begin{equation*}
        e = \dim_{\Fp}N_{K/F}(K^{\times}) > 0.
    \end{equation*}
    \item If $p=2$, $\Upsilon=1$, and $K=F(\sqrt{a})$ for a suitable
    $a\in F^{\times}\setminus F^{\times 2}$, then $-1\in N_{K/F}(K^{\times})$
    since $\Upsilon=1$.  Consequently $a\in N_{K/F}(K^{\times})$,
    and thus $1\leq e$.
\end{enumerate}
Therefore in order to prove Theorem~\ref{th:mainth} when $p>2$,
it is sufficient to show, for each cardinal numbers $d$,
$e$ as above, for each $\Upsilon\in\{0,1\}$, and for each prime
number $p>2$, the existence of a field $F$ such that:
\begin{itemize}
    \item $F$ contains a primitive $p$th root $\xi_p$;
    \item $F^\times/F^{\times p}$ decomposes intro a direct sum of
    subgroups
    \begin{equation*}
        F^\times/F^{\times p} = D\oplus\mo[a]\mc\oplus E,
    \end{equation*}
    where $\dim_{\Fp}(\mo[a]\mc \oplus E) = e$ and, setting
    $K=F(\root{p}\of{a})$,
    \begin{enumerate}
        \item $[N_{K/F}(K^{\times})]=\mo[a]\mc \oplus E$;
        \item $\dim_{\Fp}(F^{\times}/N_{K/F}(K^{\times})) =
        \dim_{\Fp} D = d$; and
        \item $\Upsilon=0$ if and only if $\xi_p\notin
        N_{K/F}(K^{\times})$.
    \end{enumerate}
\end{itemize}

In the case $p=2$ and $\Upsilon=1$ we use the same conditions as
above, and if $p=2$ and $\Upsilon=0$ we require instead that $e =
\dim_{\F_2} E$ and $d=\dim_{\F_2}(D\oplus\mo[a]\mc)$. The latter
condition is imposed because $-1\notin N_{F(\sqrt{a})/F}
(F(\sqrt{a})^{\times})$ if and only if $a\notin N_{F(\sqrt{a})/F}
(F(\sqrt{a})^{\times})$.

Our strategy is to interpret the required conditions on
$F^{\times}/ F^{\times p}$ above in terms of Galois cohomology. We
then observe that these conditions are satisfied if $G_F$ is a
free product, in the category of pro-$p$-groups, of suitable
pro-$p$-groups $G_1$ and $G_2$, and finally we use the very nice
theorem proved by Efrat and Haran which guarantees the existence
of a field with $G_F$ above. This is one of the key results used
in our paper.

\begin{theorem} \label{th:efratharan}
    (Efrat-Haran; see \cite[Proposition~1.3]{EH}) Let
    $F_1,\dots,F_n$ be fields of equal characteristic such that
    $G_{F_1},\dots,G_{F_n}$ are pro-$p$-groups. Then there exists
    a field $F$ of the same characteristic such that
    \begin{equation*}
        G_F\cong G_{F_1}\star\cdots\star G_{F_n},
    \end{equation*}
    where the product is free in the category of pro-$p$-groups.
\end{theorem}

\noindent In order to apply this theorem, we show the existence of
the fields $F_1$ and $F_2$ such that $G_{F_1}$ and $G_{F_2}$ are
prescribed Galois groups $G_1$ and $G_2$.  We use the techniques
of henselian valuations and formal power series to construct
fields $F_1$ and $F_2$.

\section{Lemmas}

\subsection{Valued fields $F_1$ with prescribed residue field
$F_0$ and valuation group $\Gamma$}\

Let $v$ be a valuation on a field $F_1$, written additively. Then
we denote by $A_v$ the valuation ring $\{f\in F_1\ \vert\ v(f)\geq
0\}$; by $M_v$ the unique maximal ideal $\{f\in A_v\ \vert\
v(f)>0\}$ of $A_v$; by $F_v$ the residue field $A_v/M_v$ of $v$;
by $\Gamma$ the valuation group $v(F_1^{\times})$ of $v$; and by
$U$ the group $A_v\setminus M_v$ of units of $v$.

The following lemma is well known and we shall omit its
straightforward proof.

\begin{lemma} \label{le:pthcount}
    Let $F_1$ be a valued field with valuation $v$, valuation
    group $\Gamma=v(F_1^\times)$, and group of units $U$.
    For each prime $p\neq \chr(F_1)$ there exists an isomorphism
    \begin{equation*}
        \varphi:F_1^{\times}/F_1^{\times p}\longrightarrow U/U^{p}
        \oplus \Gamma/p\Gamma.
    \end{equation*}
    In particular
    \begin{equation*}
        \dim_{\Fp} F_1^{\times}/F_1^{\times p} = \dim_{\Fp}
        U/U^{p} + \dim_{\Fp}\Gamma/p\Gamma.
    \end{equation*}
\end{lemma}

It is well-known that for each field $F_0$ and for each
totally ordered abelian group $\Gamma$, there exists a field $F_1$
with a valuation $v:F_1\to\Gamma\cup\{\infty\}$ such that the
residue field $F_v$ is isomorphic to $F_0$ and the valuation group
is $\Gamma$.

In order to construct such a field, set
\begin{equation*}
    F_1 = F_0((\Gamma)) :=\{f:\Gamma\to F_0\ \vert\ \supp
    (f)\text{ is well-ordered}\}.
\end{equation*}
Thus a typical element $f\in F_1$ can be written as a formal sum
$f=\sum_{g\in\Gamma} a_g t^g$ such that the set $\supp (f):=\{g\in
G\ \vert\ a_g\neq 0\}$ is a well-ordered subset of $G$. The
valuation $v$ on $f$ is defined as: $v(0)=\infty$ and $v(f)= \min
\supp (f)$ for $f\neq 0$. An important property of the valued
field $F_1$ as above is the fact that it is henselian.
(See for example \cite[(1.3)]{Rib}.)
In what follows we will identify $F_v$ with $F_0$.
We will also assume that char $F_0\neq p$.

We will be particularly interested in controlling the $p$th-power
classes of such a field.  To do so, we choose particular groups
$\Gamma$ for our valuation groups.  These groups will be direct
sums of
\begin{equation*}
    \Z_{(p)}:=\left\{\frac{a}{b}\in\Q\ \big\vert\  a,b\in\Z,b \neq
    0; \ \text{if } a\neq 0 \text{ then } (a,b)=1, p\nmid
    b\right\}.
\end{equation*}
Observe that $\Z_{(p)}$ is the valuation ring of a $p$-adic
valuation on $\Q$. Let $I$ be any non-empty, well-ordered set.
Then set
\begin{equation*}
    \Gamma=\Z_{(p)}^{(I)} := \left\{\gamma: I\to \Z_{(p)}\ \Big|
    \ \vert\supp(\gamma)\vert<\infty\right\}.
\end{equation*}
Thus $\Gamma$ is a direct sum of $\vert I\vert$ copies of
$\Z_{(p)}$. Observe that $\Z_{(p)}$ carries a natural ordering
induced from $\Q$, and then we may order $\Gamma$
lexicographically, as follows. Let
$\gamma_1\neq\gamma_2\in\Gamma$. Then $\gamma_1<\gamma_2$ if and
only if $\gamma_1(i)<\gamma_2(i)$ for the least element $i\in I$
such that $\gamma_1(i)\neq\gamma_2(i)$. Then $\Gamma$ is a
linearly ordered abelian group. Recall that each non-empty set can
be well-ordered. (See \cite[Appendix~2, Theorem~4.1]{La}.)

We choose $\Gamma$ as above because $G_{F_0((\Gamma))}$ will be
pro-$p$ (see Lemma~\ref{le:gf1prop} below) and because we may
control the $p$th-power classes with the following lemma. This
well-known lemma follows from Lemma~\ref{le:pthcount} and the fact
that the valued field $F_1$ is henselian. It is also an immediate
consequence of \cite[Lemma~1.4]{W}. Therefore we shall omit its
proof.

\begin{lemma}\label{le:pthcountf1}
    Let $F_1=F_0((\Gamma))$ as above.  Then
    \begin{align*}
        \dim_{\Fp} F_1^\times/F_1^{\times p} &= \dim_{\Fp}
        F_0^\times/F_0^{\times p} + \dim_{\Fp} \Gamma/p\Gamma
        \\ &= \dim_{\Fp} F_0^\times/F_0^{\times p} +
        \vert I\vert.
    \end{align*}
\end{lemma}

Finally, we record a criterion for $G_{F_1}$ being pro-$p$:

\begin{lemma} \label{le:gf1prop}
    Let $F_1=F_0((\Gamma))$ as above with $\chr(F_0) = 0$ and
    $G_{F_0}$ pro-$p$.  Then $G_{F_1}$ is pro-$p$ as well.
\end{lemma}

\begin{proof}
    From basic valuation theory, nicely summarized in \cite[pages~3
    and 4]{K}, and the fact that $F_1$ above is henselian,
    \begin{equation*}
        G_{F_1}\cong T\rtimes G_{F_0},
    \end{equation*}
    where the action of $G_{F_0}$ on $T$ is uniquely determined by
    the cyclotomic character mapping $G_{F_0}$ into a group of
    automorphisms of a group of roots of unity contained in
    $F_0^s$, and $T\cong\Z_{p}^{I}$, the topological
    product of $\vert I\vert$ copies of $\Z_p$. In particular, if
    $G_{F_0}$ is a pro-$p$-group, so is $G_{F_1}$.
\end{proof}

\subsection{$H^2(G_{F_1}, \Fp)$ for henselian valued fields $F_1$}\

Now we study $H^2(G_{F_1}, \Fp)$ for our henselian valued fields
$F_1$.  The next lemma, taken from \cite{W}, will be used in the
proof of Theorem~\ref{th:mainth} to show that norm groups of
cyclic $p$-extensions of $F_1$ are not too large.

Suppose that $F_1$ is a field endowed with a henselian valuation
$v$ with valuation group $\Gamma=v(F_1^\times)$. Let $F_1^{{nr}}$
denote the maximal unramified extension of $F_1$ in its separable
closure $F_1^s$. Then $G_{F_0}\cong
G_{F_1}/\Gal(F_1^s/F_1^{{nr}})$.  Therefore, after identifying
these groups, we have the inflation map
\begin{equation*}
    \inf = \inf{}_{F_0}^{F_1} : H^{*}(G_{F_0}, \Fp)
    \longrightarrow H^{*}(G_{F_1}, \Fp).
\end{equation*}
(See \cite[page~483]{W}.)

Moreover, from basic Kummer theory we have the canonical
isomorphism
\begin{equation*}
    \varphi_F : F^{\times}/F^{\times p}\longrightarrow
    H^{1}(G_F, \Fp),
\end{equation*}
as well as the corresponding canonical isomorphisms
$\varphi_{F_i}$, $i=1$, $2$.  We will denote by $(f)_F$ or
$(f_i)_{F_i}$ the images $\varphi_F([f])$ or
$\varphi_{F_i}([f_i])$.  If the context is clear we will omit the
subscript.

Assume next that $\{\pi_j,\ j\in \Jc\}$ is a set of elements of
$F_1^{\times}$ such that their images in $\Gamma/p\Gamma$ form a
basis of $\Gamma/p\Gamma$ over $\Fp$. Then we have the following
lemma, obtained as a special case of a theorem of Wadsworth.

\begin{lemma} \label{le:h2ofhens} \cite[Theorem~3.6, page~483]{W}.
    Let $F_1$ and $F_0$ be as above. Then
    \begin{align*}
        H^2(G_{F_1}, \Fp) = &\inf(H^2(G_{F_0},\Fp))\oplus_{j\in \Jc}
        \left( \inf(H^1(G_{F_0}, \Fp))\cup(\pi_j)\right) \\
        &\oplus_{\{j_1,j_2\}\subset \Jc,\ j_1\neq j_2}
        \left((\pi_{j_1})\cup(\pi_{j_2})\right).
    \end{align*}
    Moreover, for each $j\in \Jc$,
    \begin{equation*}
        \inf(H^1(G_{F_0},\Fp)) \cong \inf(H^1(G_{F_0},\Fp))
        \cup(\pi_j)
    \end{equation*}
    and for each $j_1,j_2\in \Jc$ such that $j_1\neq
    j_2$, we have $(\pi_{j_1})\cup(\pi_{j_2})\neq 0$.
\end{lemma}

Note that in the last summand of Lemma~\ref{le:h2ofhens} the sum
ranges over subsets $\{j_1,j_2\},j_1\neq j_2$ of $\Jc$ and a
choice between $(\pi_{j_{1}})\cup(\pi_{j_{2}})$ and
$(\pi_{j_{2}})\cup(\pi_{j_{1}})$ is arbitrary but fixed.

\subsection{Residue fields $F_0$ with prescribed absolute
Galois group}\

In our construction of $F$ we choose a residue field $F_0$
depending on $\Upsilon$ and $p$.  If $\Upsilon=1$ we will simply
put $F_0=\C$, but when $\Upsilon=0$ we require some special
properties of $F_0$. In particular, in order that our cyclic
extension $K=F(\root{p}\of{a})$ have the desired invariant
$\Upsilon=0$, we require that $K$ does not embed in a cyclic
Galois extension $L$ over $F$ with degree $[L:F]=p^2$, for this is
equivalent to $\xi_p\notin N_{K/F}(K^\times)$ by
\cite[Theorem~3]{A}.

To ensure that this nonembeddability condition holds, as well as
to ensure that a certain nonabelian group of order $p^3$ does not
occur as a Galois group over the field, we choose residue fields
$F_0$ with absolute Galois groups taking a special form, and it is
also convenient to require that $\vert F_0^\times/F_0^{\times
p}\vert$ is small. As it turns out, we may choose some suitable
algebraic infinite extension of $\Q$. Finitely generated
pro-$p$-absolute Galois groups over $\Q$ and more generally any
global field, were nicely classified in \cite{E2}. (See also
\cite{E1} and \cite{JP} for related results and techniques.)

The extensions we will need for $p>2$ are given in the following

\begin{lemma} \label{le:specextsofQ} \cite[page~84]{E2}
    For each prime $p>2$ there exists an algebraic extension
    $F_{0,p}$ of $\Q$ such that
    \begin{equation*}
        G_{F_{0,p}} = \mo\sigma,\tau\ \vert\
        \sigma\tau\sigma^{-1}= \tau^{p+1}\mc_{\text{pro-$p$}}
    \end{equation*}
    where the presentation is in the category of pro-$p$-groups.
\end{lemma}

Observe that the maximal abelian extension $F_{0,p}^{ab}$ of
$F_{0,p}$ has $G_{F_{0,p}}^{ab} := \Gal(F_{0,p}^{ab}/ F_{0,p})$
equal to
\begin{equation*}\label{eq:gfopab}
    G_{F_{0,p}}^{ab} \cong \mo\bar\sigma,\bar\tau\ \vert
    \ \bar\tau^{p}=1\mc=\Z_p\times\Z/p\Z
\end{equation*}
for all primes $p>2$.

\subsection{Field arithmetic and free pro-$p$ products}\

In this section we collect lemmas giving information about a field
$F$ derived from the structure of $G_F$, especially when $G_F$ is
a free pro-$p$ product of two pro-$p$ groups $G_{F_1}$ and
$G_{F_2}$.

First we record a lemma detecting the presence of primitive $p$th
roots of unity in a field $F$, based only on the structure of
$G_F$.

\begin{lemma} \label{le:pthrootexists}
    Suppose that $p>2$ and that $F$ is a field with $\chr(F)\neq
    p$ and $G_F$ pro-$p$. Then $\xi_p\in F^{\times}$.
\end{lemma}

\begin{proof}
    Because $\chr(F)\neq p$, there exists a primitive $p$th root
    $\xi_p$ of unity in $F^s$. If $\xi_p\in F^s\setminus F$ then
    $F(\xi_p)/F$ is a nontrivial Galois extension of degree
    $[F(\xi_p):F]<p$. Therefore $G_F$ has a nontrivial finite
    quotient of order coprime with $p$. This contradicts our
    assumption that $G_F$ is a pro-$p$-group. Hence $\xi_p\in
    F^{\times}$ as asserted.
\end{proof}

Now suppose that $G_F=G_{F_1}\star G_{F_2}$ for pro-$p$ absolute
Galois groups $G_F$, $G_{F_1}$, and $G_{F_2}$, where the free
product is taken in the category of pro-$p$-groups.  From
\cite[(4.3) Satz]{N} we see that the restriction homomorphism
\begin{equation}\label{eq:res}
    \res : H^{1}(G_F, \Fp)\longrightarrow H^{1}(G_{F_1},
    \Fp)\oplus H^{1}(G_{F_2}, \Fp)
\end{equation}
is an isomorphism.  Now given $(f)_F$ in $H^1(G_F, \Fp)$, we
denote the image $\res (f)_F$ by
\begin{equation*}
    \res (f)_F = (f)_{G_{F_1}} \oplus (f)_{G_{F_2}}.
\end{equation*}
This notation distinguishes, then, between $(f)_{F_1}$, which
denotes $\varphi_{F_1}(f)$ for $f\in F_1^\times$, and
$(f)_{G_{F_1}}$, which denotes the projection of $\res
\varphi_{F}(f)$ onto the first summand.

One way of interpreting this restriction map is with the following

\begin{lemma}\label{le:resinterp}
    Let $G_F=G_{F_1}\star G_{F_2}$ be pro-$p$ absolute Galois
    groups of fields containing a $p$th root of unity, and
    suppose that we have the following sequence:
    \[
    \xymatrix{ G_F\ar@{>>}[r]^{\can\ } & G_{F_1} \ar@{>>}[r] &
    \Z/p\Z. }
    \]
    Here the canonical map $\can$ is an identity on $G_{F_1}$ and
    contains $G_{F_2}$ in its kernel.

    Then the right-hand surjection and the composed surjection
    correspond to fields $K_1=F_1(\root{p}\of{a_1})$ and
    $K=F(\root{p}\of{a})$, respectively, where
    $(a)_{G_{F_1}}=(a_1)_{F_1}$ and $(a)_{G_{F_2}}=0$.
\end{lemma}

\begin{proof}
    The surjections are continuous homomorphisms, hence elements
    of $H^1(G_{F_1},\Fp)$ and $H^1(G_{F},\Fp)$, respectively, and
    the right-hand surjection is clearly the restriction of the
    composed surjection.  The remainder follows by Kummer theory.
\end{proof}

\begin{lemma} \label{le:cupprod}
    Let $G_F=G_{F_1}\star G_{F_2}$ be pro-$p$ absolute Galois
    groups and suppose that $(a)_F$ and $(b)_F$ satisfy
    $(a)_{G_{F_1}} = (b)_{G_{F_2}} = 0$.  Then
    \begin{equation*}
        (a)_F \cup (b)_F = 0 \in H^{2}(G_F, \Fp).
    \end{equation*}
\end{lemma}

The lemma follows from \cite[(4.1) Satz]{N} and from
\cite[Prop.~7.3, page~191]{Ris}. However, we prove our lemma by
translating the cup products into obstructions to basic Galois
embedding problems, yielding an interesting Galois-theoretic
variant of the proof.

\begin{proof}
    If $(a)=0$ or $(b)=0$, we are done.  Otherwise, the conditions
    $(a)_{G_{F_1}}=(b)_{G_{F_2}}=0$ imply that $(a)$ and $(b)$ are
    linearly independent in $H^1(G_F, \Fp)$.

    Now let $H_{p^3}$ be the Heisenberg group of
    order $p^3$:
    \begin{align*}
        H_{p^3} = \mo &v_1, v_2, w\ \vert\ v_1^p=v_2^p=w^p=1,
        v_2v_1=wv_1v_2, \\ &[v_1,w]=[v_2,w]=1 \mc
    \end{align*}
    In the case $p=2$, $H_{8}$ is the familiar dihedral group
    $D_4$.

    By \cite[Corollary, page 523 and Theorem 3(A)]{M}, if $(a)$
    and $(b)$ are linearly independent, then $(a)\cup (b)=0$ if
    and only if $H_{p^3}$ is the Galois group $\Gal(M/F)$ of a
    Galois extension $M$ of $F$ containing $F(\root{p}\of{a},
    \root{p}\of{b})$ in such a way that
    \begin{equation*}
        H_{p^3}/\mo v_1,w\mc = \Gal(F(\root{p}\of{a})/F)
        \text{ and }
        H_{p^3}/\mo v_2,w\mc = \Gal(F(\root{p}\of{b})/F).
    \end{equation*}

    Now consider the commutative diagram
    \[
    \xymatrix{ & & G_F\ar@{>>}[ld]_{\delta_1} \ar@{.>>}[dd]^\beta
    \ar@{>>}[rd]^{\delta_2} & & \\ & G_{F_1} \ar@{>>}[dl]_{\alpha_1}
    & & G_{F_2} \ar@{>>}[dr]^{\alpha_2} & \\ \Z/p\Z
    \ar@{^{(}->}[rr]^{1\mapsto v_1} & & H_{p^3}
    \ar@(dr,dl)[rr]_{\mo v_1, w\mc\mapsto 0; \ v_2\mapsto 1}
    \ar@(dl,dr)[ll]^{\mo v_2, w\mc\mapsto 0; \ v_1\mapsto 1} & & \Z/p\Z
    \ar@{_{(}->}[ll]_{1\mapsto v_2}
    }
    \]
    Let $a_2\in F_2^\times$ and $b_1\in F_1^\times$ satisfy
    $(a_2)_{F_2} = (a)_{G_{F_2}}$ and $(b_1)_{F_1} =
    (b)_{G_{F_1}}$.  Then set $K_1=F_1(\root{p}\of{b_1})$ and
    $K_2=F_2(\root{p}\of{a_2})$; these are $\Z/p\Z$-extensions of
    $F_1$ and $F_2$, respectively.  We may then identify the
    left-hand $\Z/p\Z$ in the diagram with $\Gal(K_1/F_1)$ so that
    $\alpha_1$ is the surjection of Galois theory.  Similarly, the
    right-hand $\Z/p\Z$ may be identified with $\Gal(K_2/F_2)$
    with $\alpha_2$ the surjection of Galois theory.  Finally,
    the topmost surjections $\delta_1$ and $\delta_2$ are canonical.

    By Lemma~\ref{le:resinterp}, the surjections
    $\delta_1\alpha_1$ and $\delta_2\alpha_2$ correspond to fields
    $F(\root{p}\of{b})$ and $F(\root{p}\of{a})$, respectively.
    Now because $G_F=G_{F_1}\star G_{F_2}$, there exists a
    homomorphism $\beta: G_F\to H_{p^3}$, and because $v_1$ and
    $v_2$ generate $H_{p^3}$, $\beta$ is a surjection.

    Hence $H_{p^3}$ is a Galois group over $F$ corresponding to a
    normal subgroup $H$ of $G_F$.  Consider the smallest normal
    subgroup $H_1$ of $G_{F}$ containing $H$ and $G_{F_2}$.  Then
    by the diagram, $G_F/H_1$ is the left-hand $\Z/p\Z$, which
    corresponds to $F(\root{p}\of{b})$, and $H_1/H$ is $\mo v_2,
    w\mc$.  Now consider the smallest normal subgroup $H_2$ of
    $G_{F}$ containing $H$ and $G_{F_1}$.  Then by the diagram,
    $G_F/H_2$ is the right-hand $\Z/p\Z$, which corresponds to
    $F(\root{p}\of{a})$, and $H_2/H$ is $\mo v_1, w\mc$.

    Hence $(a)\cup (b)=0$.
\end{proof}

Finally, we close with with a companion to
Lemma~\ref{le:resinterp}. In Lemma~\ref{le:abeloffree} below, $\pi$
denotes the canonical homomorphism of $G$ to $G_1$ which is an identity
on $G_1$, and is trivial on $G_2$.

\begin{lemma} \label{le:abeloffree}
    Let $G=G_1\star G_2$ be a free product of $G_1$ and $G_2$ in
    the category of pro-$p$-groups. Suppose that $A\cong \Z/p\Z$
    is a factor group of $G_1$ such that the surjection $G_1\to
    \Z/p\Z$ does not factor through $\Z/p^2\Z$. Then the following
    commutative diagram cannot occur:
    \[
    \xymatrix{ G\ar@{>>}[r]^-{\pi} \ar@{>>}[d] & G_1 \ar@{>>}[d] \\
    \Z/p^2\Z \ar@{>>}[r] & A. }
    \]
\end{lemma}

\begin{proof}
    Suppose that contrary to our statement, such a diagram as the
    above exists. Then by passing to quotients by commutator
    subgroups we obtain
    \[
    \xymatrix{ G^{ab} \ar@{>>}[r]^-{\pi^{ab}}
    \ar@{>>}[d]_{\alpha} &
    G_1^{ab} \ar@{>>}[d]^{\gamma} \\
    \Z/p^2\Z \ar@{>>}[r]^{\beta} & A. }
    \]
    But $G^{ab}\cong G_1^{ab}\times
    G_2^{ab}$ and the canonical surjection onto
    $G_1^{ab}$ is given by the projection map.  Let
    $\delta$ be a splitting map of the projection map.  Then
    $\gamma = \beta\alpha\delta$, contradicting the hypothesis.
\end{proof}


\section{Proof of the Theorem} \label{se:theproof}

First we define fields $F_0$, $F_1$, $F_2$, and $F$ using our
given cardinal numbers $d$, $e$, and $\Upsilon$, as well as the
prime number $p$.  Then we define the cyclic Galois extension
$K/F$ of degree $p$ and check that the arithmetic invariants of
$K/F$ coincide with $d$, $e$, and $\Upsilon$.

\subsection{Constructing $F_0$, $F_1$, $F_2$, and $F$}\

If $\Upsilon=1$ then let $F_0=\C$. If $\Upsilon=0$ and $p=2$, let
$F_0=\R$. (See Proposition~\ref{pr:trandeg} for alternative
choices in these two cases.)  If $\Upsilon=0$ and $p>2$ then let
$F_{0,p}$ be the algebraic extension of $\Q$ of
Lemma~\ref{le:specextsofQ}. In the first two cases we see
trivially that $\xi_p\in F_0$, and in the last case $\xi_p\in F_0$
by Lemma~\ref{le:pthrootexists}. Observe that
\begin{equation}\label{eq:f0pth}
        \dim_{\Fp} F_{0}^{\times}/F_{0}^{\times p}=
    \begin{cases}
        0, \text{ if } \Upsilon=1;\\
        1, \text{ if } \Upsilon=0, \ p=2;\\
        2, \text{ if } \Upsilon=0, \ p>2.
    \end{cases}
\end{equation}

We next construct the field $F_1$. Because $\Upsilon=0$ implies
$1\leq d$, for either choice of $\Upsilon\in \{0,1\}$ there exists
a well-ordered set $I_1$ such that $\vert I_1
\vert+1=d+2\Upsilon$. Let $\Gamma_1=\Z_{(p)}^{(I_1)}$ be a direct
sum of $\vert I_1 \vert$ copies of $\Z_{(p)}$. Then $\Gamma_1$ is
a linearly ordered abelian group. Finally set $F_1 :=
F_0((\Gamma_1))$. From Lemmas~\ref{le:pthcountf1} and
\ref{le:gf1prop} it follows that $G_{F_1}$ is a pro-$p$-group and
\begin{equation*}
    \dim_{\Fp}F_{1}^{\times}/F_{1}^{\times p}=
    \dim_{\Fp}F_{0}^{\times}/F_{0}^{\times p} + \vert I_1 \vert.
\end{equation*}
Hence
\begin{equation}\label{eq:f1pth}
    \dim_{\Fp} F_{1}^{\times}/F_{1}^{\times p}=
    \begin{cases}
    d+1, &\text{if } \Upsilon=1;\\
       d, &\text{if } \Upsilon=0, \ p=2;\\
       d+1, & \text{if } \Upsilon=0, \ p>2.
    \end{cases}
\end{equation}

Similarly, we construct $F_2$ as follows.  Because $p>2$ and also
$p=2$ and $\Upsilon=1$ implies
$e>0$, there exists a well-ordered set $I_2$ such that $1+\vert
I_2 \vert = e$ in either of the cases $p>2$ or $\Upsilon=1$,
$p=2$, and $\vert I_2 \vert=e$ in the case $\Upsilon=0$, $p=2$.
Then again $\Gamma_2=\Z_{(p)}^{(I_2)}$ is a linearly ordered
abelian group. We set $F_2 := \C((\Gamma_2))$. Then from
\cite[pages~3~and~4]{K} it follows that
$G_{F_2}\cong\Z_{p}^{I_2}$, the topological product of $\vert I_2
\vert$ copies of $\Z_p$. In particular
\begin{equation}\label{eq:edef}
        e =
    \begin{cases}
        \dim_{\Fp} F_{2}^{\times}/F_{2}^{\times p}+1, &\text{if }
        p>2 \text{ or } p=2 \text{ and } \Upsilon=1;\\
        \dim_{\Fp} F_{2}^{\times}/F_{2}^{\times p}, &
        \text{if }
        p=2, \ \Upsilon=0.\\
    \end{cases}
\end{equation}

From Theorem~\ref{th:efratharan} we see that there exists a field $F$ of 
characteristic zero, such
that $G_F=G_{F_1}\star G_{F_2}$ is a free product of $G_{F_1}$ and
$G_{F_2}$ in the category of pro-$p$-groups. In particular $G_F$
is again a pro-$p$-group, and from Lemma~\ref{le:pthrootexists} we
see that $F$ contains a primitive $p$th root of unity.

\subsection{Constructing $K/F$}\label{kf}\

We define the cyclic extension $K/F$ of degree $p$ as
$K=F(\root{p}\of{a})$ where $[a]\in F^{\times}/F^{\times p}$ is
chosen via the isomorphism \eqref{eq:res}.

If $\Upsilon=1$ then let $(a_1)_{F_1}$ be any nontrivial element
in $H^1(G_{F_1}, \Fp)$, which is possible since $1\leq \dim_{\Fp}
H^1(G_{F_1}, \Fp) = d+1$.  Let $(a_2)_{F_2}=0$ in $H^2(G_{F_2},
\Fp)$.

Now suppose $\Upsilon=0$ and $p>2$.  We denote the fixed field of
the factor $\Z_p$ in $G_{F_0}^{ab}\cong\Z_p\times\Z/p\Z$ acting on
a maximal abelian extension $F_{0}^{ab}$ of $F_0$ as
$K_0:=F_0(\root{p}\of{b})$. (See the discussion following
Lemma~\ref{le:specextsofQ}.) Because $F_{0}^{\times}/F_{0}^{\times
p}$ is naturally isomorphic to a subgroup of
$F_{1}^{\times}/F_{1}^{\times p}$, we may set $(a_1)_{F_1} = \inf
(b)_{F_0}\neq 0$ in $H^1(G_{F_1}, \Fp)$ and $(a_2)_{F_2}=0$ in
$H^2(G_{F_2},\Fp)$.

Finally assume that $\Upsilon=0$ and $p=2$.  We set
$(a_1)_{F_1}=(-1)_{F_1}\neq 0$ in $H^1(G_{F_1}, \F_2)$ and
$(a_2)_{F_2}=0$ in $H^2(G_{F_2},\F_2)$.

Now by \eqref{eq:res} there exists $a\in F^\times$ such that
$(a)_F\neq 0$ and $(a)_{G_{F_i}}=(a_i)_{F_i}$, $i=1$, $2$.  Since
we have already determined that $\xi_p\in F^{\times}$, we see that
$K=F(\root{p}\of{a})$ is a cyclic extension of degree $p$. Hence
it remains to show that the arithmetic invariants $d(K/F)$,
$e(K/F)$, and $\Upsilon(K/F)$ coincide with the prescribed
cardinal numbers $d$, $e$, and $\Upsilon$ respectively.

\subsection{Determining $d(K/F)$ and $e(K/F)$ via annihilators}\

We first observe some relationships among cup products of elements
in $H^1(G_F, \Fp)$.  Recall that Lemma~\ref{le:cupprod} tells us
that if $c_1, c_2\in F^\times$ with $(c_1)_{G_{F_1}} =
(c_2)_{G_{F_2}} = 0$ then
\begin{equation}\label{eq:f1f2}
    (c_1)_F \cup (c_2)_F = 0\in
    H^2(G_F, \Fp).
\end{equation}
Moreover, recall that by \cite[Satz~4.1]{N}
\begin{equation*}
    H^2(G_F,\Fp)\cong H^2(G_{F_1},\Fp)\times H^2(G_{F_2},\Fp),
\end{equation*}
where the isomorphism is induced by the restriction maps
\begin{equation*}
    \res_i : H^2(G_F,\Fp) \to H^2(G_{F_i}, \Fp), i=1, 2.
\end{equation*}
Because these restriction maps commute with cup product maps
(\cite[Proposition~7.3, page~191]{Ris}), we see that if $c_1,
c_2\in F^\times$ such that $(c_1)_{G_{F_2}}=(c_2)_{G_{F_2}}=0$ in
$H^2(G_{F_2}, \Fp)$ then
\begin{equation}\label{eq:tof1}
    (c_1)_F \cup (c_2)_F \neq 0\quad
    \text{iff}\quad (c_1)_{G_{F_1}}\cup (c_2)_{G_{F_1}}\neq 0,
\end{equation}
where the first cup product lies in $H^1(G_F, \Fp)$ and the second
cup product lies in $H^1(G_{F_1}, \Fp)$.  By symmetry the
analogous statement with $F_1$ and $F_2$ exchanged holds as well.

Now we calculate $e(K/F)$. We adopt the following notation for an
annihilator of $\phi\in H^1(G_{F},\Fp)$:
\begin{equation*}
    \ann_{F_1}\phi := \{\eta\in H^1(G_{F}, \Fp)\ \vert\
    \eta\cup\phi=0\}.
\end{equation*}
For $x\in F^{\times}$ we have $x\in
N_{K/F}(K^{\times})$ if and only if $(x)_F \cup (a)_F =0$ in
$H^2(G_F, \Fp)$.  Therefore, using \eqref{eq:f1f2} and
\eqref{eq:tof1} with the fact that in every case
$(a)_{G_{F_2}}=0$,
\begin{align*}
    e(K/F) &= \dim_{\Fp} N(K^{\times})/F^{\times p} \\
        &= \dim_{\Fp}\{(x)_F \in H^1(G_F,\Fp)\ \vert \ (a)_F
        \cup (x)_F=0\} \\
        &= \dim_{\Fp}\{(y)_{F_1} \in H^1(G_{F_1}, \Fp)\ \vert
        \ (a)_{G_{F_1}} \cup(y)_{F_1}=0\}\\
        &\phantom{=\ \ } + \dim_{\Fp}
        H^1(G_{F_2},\Fp) \\
        &= \dim_{\Fp} \ann_{F_1}(a)_{G_{F_1}} + \dim_{\Fp}
        F_2^\times/F_2^{\times p}.
\end{align*}

If $\Upsilon=1$ then $H^1(G_{F_0}, \Fp)=0$, and so by
Lemma~\ref{le:h2ofhens} we have $\dim_{\Fp} \ann_{F_1}
(a)_{G_{F_1}} = 1$.

If $p>2$ and $\Upsilon=0$ then by Lemma~\ref{le:h2ofhens},
\begin{equation*}
    \dim_{\Fp} \ann_{F_1}(a)_{G_{F_1}} = \dim_{\Fp} \ann_{F_0}
    (b)_{F_0},
\end{equation*}
where $b$ was chosen so that $K_0=F_0(\root{p}\of{b})$ was the
fixed field of the factor $\Z_p$ in $G_{F_0}^{ab}$. Now because
$(b)_{F_0} \cup (b)_{F_0} = 0$ as an identity in $H^2(G_F, \Fp)$
for $p>2$,
\begin{equation*}
    1\le \dim_{\Fp} \ann_{F_0} (b)_{F_0} \le 2.
\end{equation*}
On the other hand, let $c\in F^\times\setminus F^{\times p}$ be
defined so that $F_0(\root{p}\of{c})$ is contained in
the fixed field of the
factor $\Z/p\Z$ in $G_{F_0}^{ab}$.  Then by \eqref{eq:f0pth}, $\{
(b)_{F_0}, (c)_{F_0}\}$ spans $H^1(G_{F_0}, \Fp)$. Furthermore,
since by Lemma~\ref{le:specextsofQ}, $H_{p^3}$ is not a quotient
of $G_{F_{0,p}}$, $(b)_{F_0} \cup (c)_{F_0}\neq 0$ by
\cite[Corollary, page 523 and Theorem 3(A)]{M}. We conclude that
$\dim_{\Fp} \ann_{F_0} (b)_{F_0} = 1$.

Finally, if $p=2$ and $\Upsilon=0$ then again by
Lemma~\ref{le:h2ofhens}
\begin{equation*}
    \dim_{\F_2} \ann_{F_1}(a)_{G_{F_1}} = \dim_{\F_2}
    \ann_{F_0} (-1)_{F_0}.
\end{equation*}
As $F_0=\R$, $\dim_{\F_2} F_0^\times/F_0^{\times 2} = 1$ and
$(-1)_{\R}\cup (-1)_{\R} \neq 0$, yielding $\dim_{\F_2} \ann_{F_1}
(a)_{G_{F_1}} = 0.$

Combining our results with \eqref{eq:edef}, we have that
$e(K/F)=e$.

Now we turn to a similar calculation of $d(K/F)$.  Again using
\eqref{eq:f1f2} and \eqref{eq:tof1} with the fact that in every
case $(a)_{G_{F_2}}=0$,
\begin{align*}
    d(K/F)  &= \dim_{\Fp}(H^1(G_F,\Fp)/ \ann_{F} (a)_F)  \\
    &= \dim_{\Fp}(H^1(G_{F_1},\Fp)/\ann_{F_1} (a)_{G_{F_1}}) \\
    &= d.
\end{align*}
For this last equality we use \eqref{eq:f1pth} together with
calculations of the dimension of $\ann_{F_1} (a)_{G_{F_1}}$
already achieved. Hence $d(K/F)=d$ in all cases.

\subsection{Determining $\Upsilon$ via quotients of $G_F$}\

It remains to show that $\Upsilon(K/F)=\Upsilon$. First consider
the case $\Upsilon=1$.  Since $F_0=\C$, $\xi_p$ is a $p$th power
in $F_1$, and $F_1(\root{p}\of{a})$ embeds in a
$\Z/p^2\Z$-extension $F_1(\root{p^2}\of{a})$ of $F_1$. Then the
surjection $G_{F_1}\to\Gal(K_1/F_1)$ factors through $\Z/p^2\Z$.
Following the surjection with the canonical surjection $G_F\to
G_{F_1}$, we see that $\Z/p^2\Z$ is a factor group of $G_F$.
Moreover, by Lemma~\ref{le:resinterp}, the surjection $G_F \to
\Z/p\Z$ corresponds to $K$.  Hence $K/F$ embeds in a
$\Z/p^2\Z$-extension of $F$.  By \cite[Theorem~3]{A}, $\xi_p\in
N_{K/F}(K^{\times})$. Therefore $\Upsilon(K/F)=1=\Upsilon$.

Now consider the case $\Upsilon=0$ and $p>2$.  Because
$K_0=F_0(\root{p}\of{b})$ does not embed in a $\Z/p^2\Z$-extension
of $F_0$, $\xi_p\notin N_{K_0/F_0}(K_0^\times)$. (See \ref{kf} for
the definition of $K_0$.) Hence $(b)_{F_0} \cup(\xi_p)_{F_0}\neq
0$ in $H^1(G_{F_0},\Fp)$, and, by Lemma~\ref{le:h2ofhens}, $(\inf
(b))\cup (\xi_p)_{F_1} \neq 0$ in $H^1(G_{F_1},\Fp)$ as well.
Choose $b_1\in F_1^\times$ so that $(b_1)_{F_1} = \inf (b)_{F_0}$
and set $K_1=F_1(\root{p}\of{b_1})$. Then $\xi_p \notin
N_{K_1/F_1} (K_1^\times)$ and by \cite[Theorem~3]{A} the field
extension $K_1/F_1$ does not embed in a $\Z/p^2\Z$-extension of
$F_1$. Therefore the surjection $G_{F_1}\to
\Gal(F_1(\root{p}\of{b})/F_1)$ does not factor through $\Z/p^2\Z$.
From Lemmas~\ref{le:resinterp} and \ref{le:abeloffree} we see that
surjection $G_F\to\Gal(K/F)$ does not factor through $\Z/p^2\Z$.
Again by \cite[Theorem~3]{A} we conclude that $\xi_p\notin
N_{K/F}(K^{\times})$ and therefore $\Upsilon(K/F)=0=\Upsilon$.

Finally consider the case $\Upsilon=0$ and $p=2$. Because
$-1\notin N_{\C/\R}(\C)$ from Lemma~\ref{le:h2ofhens} we conclude
that $-1\notin N_{K_1/F_1}(K_{1}^{\times})$.  By
\cite[Theorem~3]{A} the surjection $G_{F_1}\to\Gal(K_1/F_1)$ does
not factor through $\Z/4\Z$. As in the previous case, by
Lemmas~\ref{le:resinterp} and \ref{le:abeloffree} we see that
$K/F$ does not embed in a $\Z/4\Z$-extension of $F$. Again by
\cite[Theorem~3]{A} we see that $-1\notin N_{K/F}(K^{\times})$ and
therefore $\Upsilon(K/F)=0=\Upsilon$.

Hence we have checked in all cases that $\Upsilon(K/F)=\Upsilon$,
and our proof of Theorem~\ref{th:mainth} is now complete. \qed

\begin{remark*}
In \cite[Lemma~1.2 and Proposition~1.3]{EH} Efrat and Haran
construct some fields with prescribed absolute Galois groups
together with some bounds on the transcendence degrees of these
fields. These bounds, together with the replacement of $\C$ by an
algebraic closure of $\Q$ and of $\R$ by a real-closed algebraic
number field $R$ in the case when $p=2$ and $\Upsilon=0$ in our
proof above, yield the following proposition.
\end{remark*}

\begin{proposition} \label{pr:trandeg}
Suppose that $p$ is a prime number, and let $d$, $e$, and
$\Upsilon\in\{0,1\}$ be cardinal numbers such that if $\Upsilon=0$
then $1\leq d$, if $p>2$ then $1\leq e$, and if $p=2$ and
$\Upsilon=1$ then $1\leq e$.  Then there exists a field $F$
containing $\Q(\xi_p)$ and a cyclic Galois extension $K$ of degree
$p$ over $F$ such that
\begin{equation*}
    e(K/F)=e, \quad d(K/F)=d, \quad \text{and }\quad
    \Upsilon(K/F) = \Upsilon.
\end{equation*}
Moreover
\begin{equation*}
    \trdeg (F/\Q)\leq 1+ \max\{e,d+1\}.
    \end{equation*}
In particular if $d, e\in\N\cup\{0\}$, there exists a Galois
cyclic extension $K/F$ of degree $p$ with prescribed invariants
$d$, $e$, and $\Upsilon$ of finite transcendence degree over its
prime field $\Q$.
\end{proposition}

\section{Acknowledgements} \label{S4}

We would like to thank the organizers of the MSRI programs on
Galois theory and the MSRI staff for giving us the opportunity to
meet and to begin our collaboration in the Fall of 1999. We are
also grateful to A.~Wadsworth for stimulating conversations. The
first author is very appreciative of the kind assistance of Ron
Hemphill, manager of Ivest Properties Limited (London, Canada),
for having provided excellent working conditions.

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\end{document}

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%%EndFont 
%%BeginFont: MSBM10
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%%CreationDate: 1993 Sep 17 11:10:37
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
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/FullName (MSBM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM10 def
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/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
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dup 67 /C put
dup 70 /F put
dup 78 /N put
dup 81 /Q put
dup 82 /R put
dup 90 /Z put
dup 111 /multicloseright put
readonly def
/FontBBox{-55 -420 2343 920}readonly def
/UniqueID 5031982 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMBX10
%!PS-AdobeFont-1.1: CMBX10 1.00B
%%CreationDate: 1992 Feb 19 19:54:06
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
 Reserved) readonly def
/FullName (CMBX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
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dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
readonly def
/FontBBox{-301 -250 1164 946}readonly def
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currentdict end
currentfile=
 eexec
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cleartomark
%%EndFont 
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%%RevisionDate: 1997 Sep 17 22:48:36
%
% XYBSQL10: quarter circle segments for Xy-pic at 10 point
%
% Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose.
% PostScript adaptation Copyright (C) 1994-1997 Ross Moore.
% Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc.
%
% This file is part of the Xy-pic macro package.
% Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose <krisrose@brics.dk>
%
% The Xy-pic macro package is free software; you can redistribute it
% and/or modify it under the terms of the GNU General Public License
% as published by the Free Software Foundation; either version 2
% of the License, or (at your option) any later version.
%
% The Xy-pic macro package is distributed in the hope that it will
% be useful, but WITHOUT ANY WARRANTY; without even the implied
% warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
% See the GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this macro package; if not, write to the
% Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
11 dict begin
/FontInfo 9 dict dup begin
/version (001.104) readonly def
/Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly=
 def
/FullName (XYBSQL10) readonly def
/FamilyName (XYBSQL) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
/UnderlinePosition -276 def
/UnderlineThickness 138 def
end readonly def
/FontName /XYBSQL10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 31 /d31 put
dup 63 /d63 put
dup 95 /d95 put
dup 127 /d127 put
readonly def
/FontBBox{-376 -376 376 376}readonly def
/UniqueID 5092842 def
currentdict end
currentfile=
 eexec
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cleartomark
%%EndFont 
%%BeginFont: XYDASH10
%!PS-AdobeFont-1.1: XYDASH10 001.104
%%CreationDate: 1997 Jul 20 21:19:18
%%RevisionDate: 1997 Aug 28 05:34:12
%%RevisionDate: 1997 Sep 18 10:23:31
%
% XYDASH10: line segments for Xy-pic at 10 point
%
% Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose.
% PostScript adaptation Copyright (C) 1994-1997 Ross Moore.
% Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc.
%
% This file is part of the Xy-pic macro package.
% Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose <krisrose@brics.dk>
%
% The Xy-pic macro package is free software; you can redistribute it
% and/or modify it under the terms of the GNU General Public License
% as published by the Free Software Foundation; either version 2
% of the License, or (at your option) any later version.
%
% The Xy-pic macro package is distributed in the hope that it will
% be useful, but WITHOUT ANY WARRANTY; without even the implied
% warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
% See the GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this macro package; if not, write to the
% Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
11 dict begin
/FontInfo 9 dict dup begin
/version (001.104) readonly def
/Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly=
 def
/FullName (XYDASH10) readonly def
/FamilyName (XYDASH) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
/UnderlinePosition -300 def
/UnderlineThickness 150 def
end readonly def
/FontName /XYDASH10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 68 /d68 put
dup 70 /d70 put
dup 120 /d120 put
dup 122 /d122 put
readonly def
/FontBBox{-40 -520 503 520}readonly def
/UniqueID 5092844 def
currentdict end
currentfile=
 eexec
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cleartomark
%%EndFont 
%%BeginFont: XYBTIP10
%!PS-AdobeFont-1.1: XYBTIP10 001.104
%%CreationDate: 1997 Jul 20 21:19:18
%%RevisionDate: 1997 Sep 14 19:58:47
%
% XYBTIP10: lower arrow tips for Xy-pic at 10 point "technical style".
%
% Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose.
% PostScript adaptation Copyright (C) 1994-1997 Ross Moore.
% Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc.
%
% This file is part of the Xy-pic macro package.
% Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose <krisrose@brics.dk>
%
% The Xy-pic macro package is free software; you can redistribute it
% and/or modify it under the terms of the GNU General Public License
% as published by the Free Software Foundation; either version 2
% of the License, or (at your option) any later version.
%
% The Xy-pic macro package is distributed in the hope that it will
% be useful, but WITHOUT ANY WARRANTY; without even the implied
% warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
% See the GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this macro package; if not, write to the
% Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
11 dict begin
/FontInfo 9 dict dup begin
/version (001.104) readonly def
/Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly=
 def
/FullName (XYBTIP10) readonly def
/FamilyName (XYBTIP) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
/UnderlinePosition -276 def
/UnderlineThickness 138 def
end readonly def
/FontName /XYBTIP10 def
/PaintType=200 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 15 /d15 put
dup 34 /d34 put
dup 35 /d35 put
dup 47 /d47 put
dup 56 /d56 put
dup 102 /d102 put
dup 111 /d111 put
dup 123 /d123 put
dup 124 /d124 put
readonly def
/FontBBox{-542 -542 542 542}readonly def
/UniqueID 5092839 def
currentdict end
currentfile=
 eexec
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cleartomark
%%EndFont 
%%BeginFont: XYATIP10
%!PS-AdobeFont-1.1: XYATIP10 001.104
%%CreationDate: 1997 Jul 20 21:19:17
%%RevisionDate: 1997 Sep 14 19:58:47
%
% XYATIP10: upper arrow tips for Xy-pic at 10 point "technical style".
%
% Original Metafont design Copyright (C) 1991-1997 Kristoffer H. Rose.
% PostScript adaptation Copyright (C) 1994-1997 Ross Moore.
% Hinting and ATM compatibility Copyright (C) 1997 Y&Y, Inc.
%
% This file is part of the Xy-pic macro package.
% Xy-pic Copyright (c) 1991-1997 Kristoffer H. Rose <krisrose@brics.dk>
%
% The Xy-pic macro package is free software; you can redistribute it
% and/or modify it under the terms of the GNU General Public License
% as published by the Free Software Foundation; either version 2
% of the License, or (at your option) any later version.
%
% The Xy-pic macro package is distributed in the hope that it will
% be useful, but WITHOUT ANY WARRANTY; without even the implied
% warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
% See the GNU General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this macro package; if not, write to the
% Free Software Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA.
11 dict begin
/FontInfo 9 dict dup begin
/version (001.104) readonly def
/Notice (Copyright (C) 1996, 1997 Ross Moore and Y&Y, Inc.) readonly=
 def
/FullName (XYATIP10) readonly def
/FamilyName (XYATIP) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
/UnderlinePosition -276 def
/UnderlineThickness 138 def
end readonly def
/FontName /XYATIP10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 15 /d15 put
dup 34 /d34 put
dup 35 /d35 put
dup 47 /d47 put
dup 56 /d56 put
dup 102 /d102 put
dup 111 /d111 put
dup 123 /d123 put
dup 124 /d124 put
readonly def
/FontBBox{-542 -542 542 542}readonly def
/UniqueID 5092838 def
currentdict end
currentfile eexec
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3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B
532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470
B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B
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D55493440B9008DAD1724D1025FF4C1B847BE604D73EB4978213D57EEAD8A8D1
5A86CA35DD6601510434BA8FFED4C13D902896B29BB9785C3082D736BA9823D5
7E64CA23FEA7F44B2124E5F67CEC97E9DE58B6FA5B980B36AE286B081C98D7BD
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156E71D55EC88A3532CAF855B82F85C3FE0C25A537A2341021AA624B24285243
57E0798D7E161B1C6A1E32284BA3B1236B5DA8B4FE1D35D1AC64134E2B952218
C5836BAEF2006F9DB675F519EDE9E4E20A825B8EF4636BDF1EF1BB5190CA66B0
ED8EA86545CD3632AA51A0C38C7F92B5E2538B6AC6EF9F0BD8E9C8BBE5EDB6B5
0736FC8D8A23288D93BBF0F7B3805B44B1B50985086DDF1A3CB7DEA6DBDC7ECA
CCD106F227EDEAF63D6C4CA37A6C3B32A2F582D2B33B366DD1C792DF5DCA71EA
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52685316908C9754A1FEC267FE4CF913253A16A8AC5D43E907F8DB74802C834D
861B250DC3AC668CCAA5ECD8E58D2AF9499A5B822304FA2BEBA54D325363ED21
814D5132E07D93D23D68F7366459D13762BD680FF9F4CD0F8361837F4FF9CE08
477704DF03F1C879247FA5A3FFBAFD383F35D1033E400FA06EB43DE3BE149728
E98DC609E89DC430440ACA19DF6660529E603C1F436D50169264DD4C1D4E9BF4
CB39292A174434E5623D3B09A0C314D0D80D98689B885840C74F31156CD4C5F8
C90B0C5EB79478F68008DADCC9C89E161E00975928D3C8C2118DD34033996052
BF6BEED6DFD94ED73C0E3E4BF0BD66EB5D3709D51365F8DB46B208B19E4263DF
491A5DA31CC860F6B7B6DD72C4680867A8302A5747E0BA32D71E133926C76AC6
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31AAEC2E69614E773AAE94002DF3ABEA942F737A87A6E37FB1176C0DE5B82727
5530E4D10986453F839B0D4F1D4B06911A782CE88D0191C6E5138E77E3279BA5
41FEA3C7EB434C01C878C84216A1D56B0530C7E54373E8C61E2F7D23A286322E
F961339B727A3843F054FF05A7F97B059A4F8F4DDAD08334E42D14E239B04218
2972FE5B2037A0ADC92EBD8138BEFA52E7CADD7C9C1EDD9CD72C510041278637
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3DDF1215B6DB584D993F019091E0D083CB70F61B7DA0A716705472D53F35873A
79D4A687E84582628A11E07414FCC4D4102D487F19930D0C1565B1A147C564F6
DD7E76A9777AEC1A9F126EC89E583458CE9A8CF5E112C9DE8D6DF845B80ACBAD
6C4BB4800199811060486A3310112D3B0C58833A7D9D5423F6EF43609F2FF4C4
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70BC26E6139149FA680805CF805D3F8A2DC599412DF97F426F6C66DC41764EF3
89A16F88F498D5C598ECA997C322E94A4277F5713B421C858F1691DBD6B6AF9D
EAEC80332CCA3DA1CAD95B4AC5E9D38443A2B90FAB02776C0B90AE72146B0A82
24CD887F95C0A40DC0342742E4A1C7BB660A935E817E12A64BE2DAB31123529F
61DA397B7E019DC0FCCAFAE8A2D06C7205157C902C4DCAC01D40A9286B5252E2
C65B4BA00C8C49D26ED1639FAE7D9F66C6069CD1F5DF4B6AD94428EC13B55A31
BE3915CD88D8D3DD0B4AC2E4E79D1F8CF5C0ACF195ACCE0E60C14F1914A12068
FB9C07BC759AD658FF702D9B2C21FA3D674B13143879C0F9CD3F381C1BCEC9DC
FC6644723B6FE481D911A03FC938E957B4781A6A7DF03254B0C849829EFFE45A
32E45BFA98FF13BCF5273293AB7AC733206A186057BCF284708B64D4E300FB1B
1A8B700529AB4F545E2447E07AD09AE1BC3E78B0CE9AB9CFCBFB08945048F890
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EA8A40018236CD797FCA7BE9DB9238A8FA97039AD7572CDB40505D679C213B00
68BC6BC7A79D8EE3ED185AA700704D5EEF608D64E28B206F595EF96432CB3B81
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cleartomark
%%EndFont 
%%BeginFont: CMTI8
%!PS-AdobeFont-1.1: CMTI8 1.0
%%CreationDate: 1991 Aug 18 21:07:42
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
 Reserved) readonly def
/FullName (CMTI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 45 /hyphen put
dup 111 /o put
dup 112 /p put
dup 114 /r put
readonly def
/FontBBox{-35 -250 1190 750}readonly def
/UniqueID 5000826 def
currentdict end
currentfile=
 eexec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4482FEDF673EDED38EF3173C475C34BFB3F6623C5E942A7797FEDDD0EF1D54E9
1D90D7076C0A9687E334907C22F2E7C603388D8D626B0E5A62B7543DCBB575D7
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BB33455799E8120D2A4862424AA4A382972E2845E042506FC8F6E201D11DD0F2
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0993A846ABF5CE645CA7CA30F82D078EA1613289A49B145835C152F4A716D96C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cleartomark
%%EndFont 
%%BeginFont: MSAM10
%!PS-AdobeFont-1.1: MSAM10 2.1
%%CreationDate: 1993 Sep 17 09:05:00
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
 Reserved) readonly def
/FullName (MSAM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSAM10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 3 /square put
readonly def
/FontBBox{8 -463 1331 1003}readonly def
/UniqueID 5031981 def
currentdict end
currentfile=
 eexec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cleartomark
%%EndFont 
%%BeginFont: CMCSC10
%!PS-AdobeFont-1.1: CMCSC10 1.0
%%CreationDate: 1991 Aug 18 17:46:49
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
 Reserved) readonly def
/FullName (CMCSC10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMCSC10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 48 /zero put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 87 /W put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
readonly def
/FontBBox{14 -250 1077 750}readonly def
/UniqueID 5000772 def
currentdict end
currentfile eexec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017B3ACC60ABE3878E1CFBC49927F065AA44C9AE92A862F55E1E7309E51D62F7
9D201C1F590B76A203820B391F397BFE82B7E0A938C07CBFF8DBEC7D835AD76D
48C1E5C986BA86634A8DA89EA8661ACC44B603CD5EDFF7A10B757A07AB55F625
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cleartomark
%%EndFont 
%%BeginFont: EUFM10
%!PS-AdobeFont-1.1: EUFM10 2.1
%%CreationDate: 1992 Nov 20 17:36:20
% Euler fonts were designed by Hermann Zapf.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
 Reserved) readonly def
/FullName (EUFM10) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /EUFM10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 75 /K put
readonly def
/FontBBox{-26 -224 1055 741}readonly def
/UniqueID 5031986 def
currentdict end
currentfile eexec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4A6F13E41191AA13358515D902E653B81AEF60FE07C4AB673435504C2C376BB2
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B2EDAB832BCBEDA30B2472AE7663AFDFA77BC44D5E5FA73E5EDAF33A9FB0B83C
3A30CC2651486DBC7E96AB58BFD12FFD08D45DCC1F02541F6F150366A19911D5
12C3B3704B1F0616BADE7D628DBC8D96025767A6A9B484038EF61F85D5EA6217
6751D8F1E5DD680A98B0B3FB5A1F43B85E8DDE3DBF4AE96455730F29C84F7800
002BAB3A7ECDBD4EDCCD07E66DC476CFD69E64A8D0D48F4EB1BAB56007D43B79
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0A6C93B3ECAC8B6A446C18BAD1C58B96A519C79A0F942B96A2377FA521F163DF
558BD7CF586B6FDA403923105236977DED0A00EDD142B008E6AE370FD5EA504D
FD8B39EAA9FBBC85AD96B3E711E5147178F779776A7847097E5DB54F1A2EB038
B2D8C5A5137369A76337F248712236C3BFC80E7C221485F052A027F2709CC31F
12D5EB0FBF45ABE272DCA2D8D74A382918545CF9B7435456DCE365689D73CA06
2B
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cleartomark
%%EndFont 
%%BeginFont: CMR8
%!PS-AdobeFont-1.1: CMR8 1.0
%%CreationDate: 1991 Aug 20 16:39:40
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
 Reserved) readonly def
/FullName (CMR8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /Gamma put
dup 40 /parenleft put
dup 41 /parenright put
dup 43 /plus put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 56 /eight put
dup 59 /semicolon put
dup 61 /equal put
dup 97 /a put
dup 99 /c put
dup 110 /n put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
/UniqueID 5000791 def
currentdict end
currentfile=
 eexec
D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891
016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171
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D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758
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639CF3725DCAA1523DABA421C5EBA371DEB1B0BAA9AA9BE945C9FA058BB14D0A
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981F9E6AEF68E18CAC40539193DF97CDC22DAA366839E6210AC802E4D06FE2C4
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2D8595C321829FA4964D5CCBA532221607150C457C9C45528842BE52B33F3661
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CAF17B485329246A5703C9D7A7202C0E5B05291D0D49610A50191BE90ED1D353
3055897966B223519CEF08430E60B8A3DF11EF0D21A201EAA80884F44F3DF873
594AAEEC9FD18876E81713B6A311C3243BDBEB6710EDCB7C8A948E4EB9E1BA4C
DFC1500F027F470BE71356C1DADF19764337FEF1EAD741CE9E36BD7AA34DCC5B
F76A2B0A616F11C8301D1D17BAEAFC67E22A9B961C7E9F66F39BA679725D0AF1
6DE54C71B8F13591DCD5737F232D1B972B14522A6588083B656E38290AA8F5B9
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00094689CECD08E9A90A0378C94043D49241E6640626A1A9FE42097CC802C17F
CC1049FCF5EA2A4F10649B7F09D1A4E803F726406A335F2D81761F5778D1088C
1AE9AEDD31F16893C3FF75C7223362ED568977BD07AB2A92A35502E0E2C1EF6F
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334923319DBFF4FDFFD2EEBF2041088374CAC0EBEE41E135D615801D1B96C3B4
9DB9472E5615BF8CDC267A2F0CA11730E8D806AA8A3291DF0357360B93D85A9E
7F7BE196E78741CC5A66103689774CAA36BF49CCCD39EC0F9FDC954055E383F4
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C494E3EF8A916228F77FB8C9F694EF7C9C9FD887486B63503F840EA580AE7356
A9EEF259415A1BABD5B15EFAF7C53C42E07E6A044199B74724E58DA57D3B1EB3
73E39AA7223FBD102E55BED12437B7F007D99EA3E84587D7F8D5942C256C3AB9
99D4FBEBDBDDDAF969EC37EE9C04FCEC8FEA27D401A67FEA68E45418660AABF5
1B20D4CA342FE04178836A5456ED80CE00D7182A6F1F49166E6B9125C416E692
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cleartomark
%%EndFont 
%%BeginFont: CMR6
%!PS-AdobeFont-1.1: CMR6 1.0
%%CreationDate: 1991 Aug 20 16:39:02
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
 Reserved) readonly def
/FullName (CMR6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
readonly def
/FontBBox{-20 -250 1193 750}readonly def
/UniqueID 5000789 def
currentdict end
currentfile=
 eexec
D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891
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cleartomark
%%EndFont 
%%BeginFont: EUFM7
%!PS-AdobeFont-1.1: EUFM7 2.1
%%CreationDate: 1992 Nov 20 17:36:25
% Euler fonts were designed by Hermann Zapf.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
 Reserved) readonly def
/FullName (EUFM7) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /EUFM7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 75 /K put
readonly def
/FontBBox{0 -250 1193 750}readonly def
/UniqueID 5031992 def
currentdict end
currentfile=
 eexec
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cleartomark
%%EndFont 
%%BeginFont: CMEX10
%!PS-AdobeFont-1.1: CMEX10 1.00
%%CreationDate: 1992 Jul 23 21:22:48
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
 Reserved) readonly def
/FullName (CMEX10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMEX10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /parenleftbig put
dup 1 /parenrightbig put
dup 12 /vextendsingle put
dup 32 /parenleftBigg put
dup 33 /parenrightBigg put
dup 40 /braceleftBigg put
dup 48 /parenlefttp put
dup 49 /parenrighttp put
dup 56 /bracelefttp put
dup 58 /braceleftbt put
dup 60 /braceleftmid put
dup 62 /braceex put
dup 64 /parenleftbt put
dup 65 /parenrightbt put
dup 77 /circleplusdisplay put
dup 80 /summationtext put
dup 110 /braceleftBig put
dup 111 /bracerightBig put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
/UniqueID 5000774 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMTI12
%!PS-AdobeFont-1.1: CMTI12 1.0
%%CreationDate: 1991 Aug 18 21:06:53
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
 Reserved) readonly def
/FullName (CMTI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 12 /fi put
dup 40 /parenleft put
dup 41 /parenright put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 58 /colon put
dup 59 /semicolon put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 80 /P put
dup 83 /S put
dup 84 /T put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-36 -251 1103 750}readonly def
/UniqueID 5000829 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR9
%!PS-AdobeFont-1.1: CMR9 1.0
%%CreationDate: 1991 Aug 20 16:39:59
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
 Reserved) readonly def
/FullName (CMR9) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR9 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 19 /acute put
dup 20 /caron put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 65 /A put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 74 /J put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 87 /W put
readonly def
/FontBBox{-39 -250 1036 750}readonly def
/UniqueID 5000792 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMTI10
%!PS-AdobeFont-1.1: CMTI10 1.00B
%%CreationDate: 1992 Feb 19 19:56:16
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
 Reserved) readonly def
/FullName (CMTI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 11 /ff put
dup 12 /fi put
dup 19 /acute put
dup 39 /quoteright put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 58 /colon put
dup 65 /A put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 75 /K put
dup 76 /L put
dup 79 /O put
dup 80 /P put
dup 82 /R put
dup 83 /S put
dup 87 /W put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-163 -250 1146 969}readonly def
/UniqueID 5000828 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI6
%!PS-AdobeFont-1.1: CMMI6 1.100
%%CreationDate: 1996 Jul 23 07:53:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
 Reserved) readonly def
/FullName (CMMI6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI6 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 59 /comma put
dup 70 /F put
dup 97 /a put
dup 98 /b put
dup 105 /i put
dup 110 /n put
dup 112 /p put
readonly def
/FontBBox{11 -250 1241 750}readonly def
/UniqueID 5087381 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: MSBM7
%!PS-AdobeFont-1.1: MSBM7 2.1
%%CreationDate: 1992 Oct 17 08:30:50
% Math Symbol fonts were designed by the American Mathematical Society.
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (2.1) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
 Reserved) readonly def
/FullName (MSBM7) readonly def
/FamilyName (Euler) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /MSBM7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 67 /C put
dup 70 /F put
dup 82 /R put
readonly def
/FontBBox{0 -504 2615 1004}readonly def
/UniqueID 5032014 def
currentdict end
currentfile=
 eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY10
%!PS-AdobeFont-1.1: CMSY10 1.0
%%CreationDate: 1991 Aug 15 07:20:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
 Reserved) readonly def
/FullName (CMSY10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 1 /periodcentered put
dup 2 /multiply put
dup 8 /circleplus put
dup 15 /bullet put
dup 20 /lessequal put
dup 21 /greaterequal put
dup 24 /similar put
dup 33 /arrowright put
dup 49 /infinity put
dup 50 /element put
dup 54 /negationslash put
dup 74 /J put
dup 91 /union put
dup 102 /braceleft put
dup 103 /braceright put
dup 104 /angbracketleft put
dup 105 /angbracketright put
dup 106 /bar put
dup 110 /backslash put
dup 112 /radical put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
/UniqueID 5000820 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMSY8
%!PS-AdobeFont-1.1: CMSY8 1.0
%%CreationDate: 1991 Aug 15 07:22:10
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
 Reserved) readonly def
/FullName (CMSY8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /minus put
dup 2 /multiply put
dup 3 /asteriskmath put
dup 26 /propersubset put
dup 33 /arrowright put
dup 50 /element put
dup 54 /negationslash put
dup 55 /mapsto put
dup 74 /J put
dup 102 /braceleft put
dup 103 /braceright put
dup 104 /angbracketleft put
dup 105 /angbracketright put
dup 112 /radical put
readonly def
/FontBBox{-30 -955 1185 779}readonly def
/UniqueID 5000818 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI8
%!PS-AdobeFont-1.1: CMMI8 1.100
%%CreationDate: 1996 Jul 23 07:53:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
 Reserved) readonly def
/FullName (CMMI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI8 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 11 /alpha put
dup 12 /beta put
dup 13 /gamma put
dup 14 /delta put
dup 25 /pi put
dup 59 /comma put
dup 61 /slash put
dup 70 /F put
dup 71 /G put
dup 73 /I put
dup 75 /K put
dup 97 /a put
dup 98 /b put
dup 103 /g put
dup 105 /i put
dup 106 /j put
dup 110 /n put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 118 /v put
dup 119 /w put
readonly def
/FontBBox{-24 -250 1110 750}readonly def
/UniqueID 5087383 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI12
%!PS-AdobeFont-1.1: CMMI12 1.100
%%CreationDate: 1996 Jul 27 08:57:55
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
 Reserved) readonly def
/FullName (CMMI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 11 /alpha put
dup 12 /beta put
dup 13 /gamma put
dup 14 /delta put
dup 17 /eta put
dup 24 /xi put
dup 25 /pi put
dup 27 /sigma put
dup 28 /tau put
dup 30 /phi put
dup 39 /phi1 put
dup 58 /period put
dup 59 /comma put
dup 60 /less put
dup 61 /slash put
dup 62 /greater put
dup 63 /star put
dup 65 /A put
dup 66 /B put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 74 /J put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 82 /R put
dup 84 /T put
dup 85 /U put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 105 /i put
dup 106 /j put
dup 112 /p put
dup 116 /t put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
readonly def
/FontBBox{-30 -250 1026 750}readonly def
/UniqueID 5087386 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMR12
%!PS-AdobeFont-1.1: CMR12 1.0
%%CreationDate: 1991 Aug 20 16:38:05
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
 Reserved) readonly def
/FullName (CMR12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding=20256 array
0 1 255 {1 index exch /.notdef put} for
dup 0 /Gamma put
dup 7 /Upsilon put
dup 11 /ff put
dup 12 /fi put
dup 13 /fl put
dup 14 /ffi put
dup 22 /macron put
dup 40 /parenleft put
dup 41 /parenright put
dup 43 /plus put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 58 /colon put
dup 59 /semicolon put
dup 61 /equal put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 74 /J put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 87 /W put
dup 91 /bracketleft put
dup 93 /bracketright put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 106 /j put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-34 -251 988 750}readonly def
/UniqueID 5000794 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI7
%!PS-AdobeFont-1.1: CMMI7 1.100
%%CreationDate: 1996 Jul 23 07:53:53
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
 Reserved) readonly def
/FullName (CMMI7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 112 /p put
readonly def
/FontBBox{0 -250 1171 750}readonly def
/UniqueID 5087382 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMMI10
%!PS-AdobeFont-1.1: CMMI10 1.100
%%CreationDate: 1996 Jul 23 07:53:57
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.100) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
 Reserved) readonly def
/FullName (CMMI10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 61 /slash put
dup 70 /F put
dup 75 /K put
dup 112 /p put
readonly def
/FontBBox{-32 -250 1048 750}readonly def
/UniqueID 5087385 def
currentdict end
currentfile=
 eexec
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0EE709EFC27533B5A1B3B9A5707353FA04E96D96F34450A214C54B58B243225E
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cleartomark
%%EndFont 
%%BeginFont: CMSY7
%!PS-AdobeFont-1.1: CMSY7 1.0
%%CreationDate: 1991 Aug 15 07:21:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
 Reserved) readonly def
/FullName (CMSY7) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.035 def
/isFixedPitch false def
end readonly def
/FontName /CMSY7 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 2 /multiply put
dup 3 /asteriskmath put
dup 121 /dagger put
dup 122 /daggerdbl put
readonly def
/FontBBox{-15 -951 1252 782}readonly def
/UniqueID 5000817 def
currentdict end
currentfile eexec
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A213ACB58AA0A658908035BF2ED8531779838A960DFE2B27EA49C37156989C85
E21B3ABF72E39A89232CD9F4237FC80C9E64E8425AA3BEF7DED60B122A52922A
221A37D9A807DD01161779DDE7D251491EBF65A98C9FE2B1CF8D725A70281949
8F4AFFE638BBA6B12386C7F32BA350D62EA218D5B24EE612C2C20F43CD3BFD0D
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cleartomark
%%EndFont 
%%BeginFont: CMR10
%!PS-AdobeFont-1.1: CMR10 1.00B
%%CreationDate: 1992 Feb 19 19:54:52
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.00B) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
 Reserved) readonly def
/FullName (CMR10) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR10 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 12 /fi put
dup 19 /acute put
dup 20 /caron put
dup 39 /quoteright put
dup 40 /parenleft put
dup 41 /parenright put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 48 /zero put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 58 /colon put
dup 65 /A put
dup 66 /B put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 74 /J put
dup 75 /K put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 81 /Q put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 86 /V put
dup 87 /W put
dup 89 /Y put
dup 91 /bracketleft put
dup 93 /bracketright put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 120 /x put
dup 121 /y put
dup 123 /endash put
readonly def
/FontBBox{-251 -250 1009 969}readonly def
/UniqueID 5000793 def
currentdict end
currentfile eexec
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cleartomark
%%EndFont 
%%BeginFont: CMBX12
%!PS-AdobeFont-1.1: CMBX12 1.0
%%CreationDate: 1991 Aug 20 16:34:54
% Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
11 dict begin
/FontInfo 7 dict dup begin
/version (1.0) readonly def
/Notice (Copyright (C) 1997 American Mathematical Society. All Rights=
 Reserved) readonly def
/FullName (CMBX12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Bold) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMBX12 def
/PaintType 0 def
/FontType 1 def
/FontMatrix [0.001 0 0 0.001 0 0] readonly def
/Encoding 256 array
0 1 255 {1 index exch /.notdef put} for
dup 12 /fi put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 49 /one put
dup 50 /two put
dup 51 /three put
dup 52 /four put
dup 53 /five put
dup 54 /six put
dup 55 /seven put
dup 56 /eight put
dup 57 /nine put
dup 65 /A put
dup 67 /C put
dup 68 /D put
dup 69 /E put
dup 70 /F put
dup 71 /G put
dup 73 /I put
dup 76 /L put
dup 77 /M put
dup 78 /N put
dup 79 /O put
dup 80 /P put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 86 /V put
dup 97 /a put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 101 /e put
dup 102 /f put
dup 103 /g put
dup 104 /h put
dup 105 /i put
dup 107 /k put
dup 108 /l put
dup 109 /m put
dup 110 /n put
dup 111 /o put
dup 112 /p put
dup 113 /q put
dup 114 /r put
dup 115 /s put
dup 116 /t put
dup 117 /u put
dup 118 /v put
dup 119 /w put
dup 121 /y put
readonly def
/FontBBox{-53 -251 1139 750}readonly def
/UniqueID 5000769 def
currentdict end
currentfile eexec
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1 0 bop 538 716 a FF(CONSTR)m(UCTION)47 b(AND)h(CLASSIFICA)-9
b(TION)48 b(OF)h(SOME)1429 832 y(GALOIS)g(MODULES)1186
1081 y FE(J)1239 1060 y(\023)1229 1081 y(AN)28 b(MIN)1559
1060 y(\023)1549 1081 y(A)1620 1060 y(\024)1611 1081
y(C)1671 1051 y FD(\003y)1768 1081 y FE(AND)h(JOHN)e(SW)-9
b(ALLO)n(W)2679 1051 y FD(z)754 1405 y FC(Abstra)n(ct.)42
b FE(In)27=
 b(our)g(previous)g(pap)r(er)g(w)n(e)h(describ)r(e)f(the)h
(Galois)f(mo)r(dule)754 1505 y(structures)37 b(of)h=
 FB(p)p
FE(th-p)r(o)n(w)n(er)e(class)h(groups)f FB(K)2218 1475
y FD(\002)2273 1505 y FB(=3DK)2392 1475 y FD(\002)p FA(p)2481
1505 y FE(,)41 b(where)c FB(K)q(=3DF)49 b FE(is)37 b(a)754
1604 y(cyclic)c(extension)f(of)g(degree)f FB(p)i=
 FE(o)n(v)n(er)d(a)i
(\014eld)h FB(F)44 b FE(con)n(taining)32 b(a)g(primitiv)n(e)754
1704 y FB(p)p FE(th)23 b(ro)r(ot)e(of)h(unit)n(y)-7 b(.)36
b(Our)21=
 b(description)h(relies)f(up)r(on)i(arithmetic)f(in)n(v)-5
b(arian)n(ts)754 1804 y(asso)r(ciated)19 b(with)i FB(K)q(=3DF)12
b FE(.)34 b(Here)19=
 b(w)n(e)h(construct)g(\014eld)g(extensions)g
FB(K)q(=3DF)31 b FE(with)754 1903 y(prescrib)r(ed)23=
 b(arithmetic)g(in)n
(v)-5 b(arian)n(ts,)22 b(th)n(us)h(completing)g(our)f(classi\014cation)
754 2003 y(of)28 b(Galois)f(mo)r(dules)g FB(K)1506 1973
y FD(\002)1562 2003 y FB(=3DK)1681 1973 y FD(\002)p FA(p)1770
2003 y FE(.)555 2417 y Fz(Let)41 b Fy(F)55 b=
 Fz(b)s(e)41
b(a)g(\014eld)h(of)e(c)m(haracteristic)j(not)e=
 Fy(p)f
Fz(con)m(taining)i(a)f(primitiv)m(e)h Fy(p)p Fz(th)456
2533 y(ro)s(ot)37 b(of)i(unit)m(y)h Fy(\030)1092 2548
y Fx(p)1131 2533 y Fz(.)62 b(F)-8 b(or)38=
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(extension)i Fy(K)r(=3DF)52 b Fz(with)39 b(Galois)g(group)456
2649 y(Gal)o(\()p Fy(K)r(=3DF)14 b Fz(\))39 b(of)g(order)h
Fy(p)p Fz(,)i(let)f Fy(J)49 b Fz(=3D)40 b Fy(K)1890 2613
y Fw(\002)1949 2649 y Fy(=3DK)2088 2613 y Fw(\002)p Fx(p)2183
2649 y Fz(,)h(and)f(let)h Fy(N)50 b Fz(denote)41 b(the)f(norm)456
2765 y(map)32 b(from)h Fy(K)39 b Fz(to)33 b Fy(F)14 b
Fz(.)555 2975 y(In)37=
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(structure)g(of)f(the)h Fv(F)2568 2990 y Fx(p)2608 2975
y Fz([Gal)o(\()p Fy(K)r(=3DF)14 b Fz(\)]-mo)s(dule)456
3091 y Fy(J)41 b Fz(is)33=
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(three)g(arithmetic)h(in)m(v)-5 b(arian)m(ts:)679 3334
y Fu(\017)41 b Fy(d)28 b Fz(=3D)f Fy(d)p Fz(\()p Fy(K)r(=3DF)14
b Fz(\))27 b(:=3D)g(dim)1610 3349 y Ft(F)1651 3357 y Fs(p)1707
3334 y Fy(F)1784 3298 y Fw(\002)1843 3334 y Fy(=3DN)10
b Fz(\()p Fy(K)2108 3298 y Fw(\002)2167 3334 y Fz(\),)679
3451 y Fu(\017)41 b Fy(e)28 b Fz(=3D)g Fy(e)p Fz(\()p Fy(K)r(=3DF)14
b Fz(\))27 b(:=3D)g(dim)1599 3466 y Ft(F)1640 3474 y Fs(p)1696
3451 y Fy(N)10 b Fz(\()p Fy(K)1912 3415 y Fw(\002)1972
3451 y Fz(\))p Fy(=3DF)2136 3415 y Fw(\002)p Fx(p)2230
3451 y Fz(,)33 b(and)679 3569 y Fu(\017)41 b Fz(\007\()p
Fy(K)r(=3DF)14 b Fz(\))27 b(:=3D)32=
 b(1)h(or)f(0)g(according)h(to)f
(whether)i Fy(\030)2552 3584 y Fx(p)2619 3569 y Fu(2)28
b Fy(N)10 b Fz(\()p Fy(K)2929 3533 y Fw(\002)2989 3569
y Fz(\))32 b(or)h(not.)555 3811 y(No)m(w)38 b(if)g Fy(G)e
Fz(=3D)g Fv(Z)p Fy(=3Dp)p Fv(Z)p Fz(,)k(then)e Fy(J)46 b
Fz(ma)m(y)39 b(b)s(e)f(considered)h(an)f Fv(F)2765 3826
y Fx(p)2804 3811 y Fz([)p Fy(G)p Fz(]-mo)s(dule)g(via)456
3927 y(an)m(y)43 b(isomorphism)i Fy(G)1354 3900 y Fu(\030)1355
3932 y Fz(=3D)1477 3927 y(Gal\()p Fy(K)r(=3DF)14 b=
 Fz(\),)45
b(and)e(the)h(mo)s(dule)f(structure)i(of)e Fy(J)52 b
Fz(is)456 4044 y(indep)s(enden)m(t)29=
 b(of)e(the)h(c)m(hoice)h(of)e
(isomorphism.)43 b(It)28 b(is)g(a)f(fundamen)m(tal)i(problem)456
4160 y(to)41 b(classify)j(the)e(isomorphism)i(classes)f(of)f(mo)s
(dules)h Fy(J)51 b Fz(for)41 b(all)h Fy(K)r(=3DF)55 b Fz(in)42
b(our)456 4276 y(con)m(text.)h(This)30=
 b(problem)f(is)h(solv)m(ed)g(in)
f(Theorem)h(1)e(b)s(elo)m(w.)43 b(Corollaries)30 b(1)e(and)456
4392 y(2)k(of)g(Theorem)i(1)e(describ)s(e)i(all)f(mo)s(dules)g
Fy(J)42 b Fz(in)33 b(an)f(explicit)j(w)m(a)m(y)-8 b(.)p
456 4526 499 4 v 555 4620 a Fr(Date)6 b FE(:)28 b(April)g(4,)f(2003.)
555 4687 y FD(\003)593 4717 y=
 FE(Researc)n(h)g(supp)r(orted)h(in)g
(part)f(b)n(y)h(the)g(Natural)f(Sciences)h(and)g(Engineering)e(Researc)
n(h)456 4817=
 y(Council)k(of)h(Canada,)g(and)g(b)n(y)f(the)i(sp)r(ecial)
e(Dean)h(of)g(Science)g(F)-7 b(und)31=
 b(at)g(the)h(Univ)n(ersit)n(y)e
(of)456 4917 y(W)-7 b(estern)27 b(On)n(tario.)555 4986
y FD(y)590 5016 y=
 FE(Supp)r(orted)h(b)n(y)f(the)h(Mathematical)f
(Sciences)g(Researc)n(h)f(Institute,)j(Berk)n(eley)-7
b(.)555 5086 y FD(z)590 5116 y FE(Researc)n(h)27=
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(part)e(b)n(y)h(National)g(Securit)n(y)f(Agency)h(gran)n(t)f(MD)n
(A904-02-1-)456 5216 y(0061.)1931 5315 y Fq(1)p eop
%%Page: 2 2
2 1 bop 456 255 a Fq(2)797 b(J)1340 236 y(\023)1330 255
y(AN)26 b(MIN)1638 236 y(\023)1628 255 y(A)1695 236 y(\024)1686
255 y(C)f(AND)f(JOHN)h(SW)-9 b(ALLO)n(W)555 450 y=
 Fz(In)29
b(the)f(follo)m(wing)h(theorem)g(w)m(e)g(determine)h(the)f(sets)g(of)f
(in)m(v)-5 b(arian)m(ts)29 b(\()p Fy(d;)17 b(e;)g Fz(\007\))456
566 y(whic)m(h)39 b(ma)m(y)h(b)s(e)e(realized)i(b)m(y)f(an)f(extension)
i Fy(K)r(=3DF)52 b Fz(and)38 b(in)h(so)f(doing)h(classify)456
683 y(all)32 b Fv(F)652 698 y Fx(p)692 683 y Fz([)p Fy(G)p
Fz(]-mo)s(dules)h Fy(J)k Fz(=3D)27 b Fy(J)9 b Fz(\()p Fy(K)r(=3DF)14
b Fz(\))32 b(up)h(to)f(isomorphism.)456 859 y FF(Theorem)g(1.)37
b Fp(L)-5 b(et)31 b Fy(p)f Fp(b)-5 b(e)29 b(a)h(prime)g(numb)-5
b(er.)42 b(F)-7 b(or)30 b(arbitr)-5 b(ary)30 b(c)-5=
 b(ar)g(dinal)29
b(numb)-5 b(ers)456 975 y Fy(d)p Fp(,)40 b Fy(e)p=
 Fp(,)i(and)d(for)h
Fz(\007)d Fu(2)h(f)p Fz(0)p Fy(;)17 b Fz(1)p Fu(g)p Fp(,)40
b(ther)-5 b(e)40 b(exists)g(a)f(cyclic)h(\014eld)f(extension)g
Fy(K)r(=3DF)54 b Fp(of)456 1092 y(de)-5 b(gr)g(e)g(e)25
b Fy(p)g Fp(c)-5 b(ontaining)25 b(a)g(primitive)g Fy(p)p
Fp(th)h(r)-5 b(o)g(ot)25 b(of)h(unity)g(with)g(invariants)e
Fz(\()p Fy(d;)17 b(e;)g Fz(\007\))456 1208 y Fp(if)34
b(and)h(only)f(if)679 1431 y Fu(\017)41 b Fp(if)35 b
Fz(\007)27 b(=3D)h(0)p Fp(,)34 b(then)h Fz(1)28 b Fu(\024)g
Fy(d)p Fp(,)679 1547 y Fu(\017)41 b Fp(if)35 b Fy(p)27
b(>)h Fz(2)35 b Fp(then)f Fz(1)28 b Fu(\024)g Fy(e)p
Fp(,)35 b(and)679 1663 y Fu(\017)41 b Fp(if)35 b Fy(p)27
b Fz(=3D)h(2)35 b Fp(and)f Fz(\007)27 b(=3D)h(1)35 b Fp(then)f
Fz(1)28 b Fu(\024)g Fy(e)p Fp(.)555 1925 y Fz(F)-8=
 b(rom)29
b(the)i(theorem)f(ab)s(o)m(v)m(e)h(and)f(from)f([MS)q(,)h(Theorem)h(3)f
(and)g(Corollary)g(2])456 2041=
 y(w)m(e)e(immediately)g(obtain)f(the)g
(follo)m(wing)h(corollaries.)42 b(W)-8 b(e)27 b(denote)h(b)m(y)g
Fy(M)3202 2056 y Fx(i;j)3309 2041 y Fz(the)456 2157 y
Fy(j)6 b Fz(th)30 b(cyclic)j(mo)s(dule)e Fv(F)1294 2172
y Fx(p)1334 2157 y Fz([)p Fy(G)p Fz(])g(suc)m(h)h(that)e(dim)2086
2172 y Ft(F)2127 2180 y Fs(p)2184 2157 y Fy(M)2278 2172
y Fx(i;j)2386 2157 y Fz(=3D)d Fy(i)p Fz(,)32 b(where)g
Fy(j)k Fz(is)c(a)e(suitable)456 2273 y(index.)456 2450
y FF(Corollary)i(1.)37 b Fp(L)-5 b(et)29 b Fy(p)f(>)f
Fz(2)i Fp(b)-5 b(e)29 b(a)g(prime)f(numb)-5 b(er,)30
b(and)f(let)g Fy(G)g Fp(b)-5 b(e)29 b(a)g(cyclic)g(gr)-5
b(oup)456 2566 y(of)33 b(or)-5 b(der)33 b Fy(p)p Fp(.)45
b(Then)33 b(an)g Fv(F)1400 2581 y Fx(p)1440 2566 y Fz([)p
Fy(G)p Fz(])p Fp(-mo)-5 b(dule)33 b Fy(J)43 b Fp(is)33
b(r)-5 b(e)g(alizable)33 b(as)g(an)g Fv(F)2877 2581 y
Fx(p)2917 2566 y Fz([Gal)o(\()p Fy(K)r(=3DF)14 b Fz(\)])p
Fp(-)456 2682 y(mo)-5 b(dule)40 b Fy(K)883 2646 y Fw(\002)942
2682 y Fy(=3DK)1081 2646 y Fw(\002)p Fx(p)1217 2682 y=
 Fp(for)h(some)g
(cyclic)g Fy(G)p Fp(-extension)f Fy(K)r(=3DF)54 b Fp(such)41
b(that)h Fy(F)55 b Fp(c)-5 b(on-)456 2799 y(tains)43
b(a)h(primitive)g Fy(p)p Fp(th)g(r)-5 b(o)g(ot)44=
 b(of)g(unity)h(if)f
(and)g(only)f(if)h(ther)-5 b(e)44 b(exist)g(c)-5 b(ar)g(dinal)456
2915 y(numb)g(ers)34 b Fy(d)p Fp(,)g Fy(e)p Fp(,)h(and)g
Fz(\007)27 b Fu(2)h(f)p Fz(0)p Fy(;)17 b Fz(1)p Fu(g)34
b Fp(such)h(that)620 3138 y Fz(\()p Fy(i)p Fz(\))41 b
Fp(If)34 b Fz(\007)28 b(=3D)g(0)p Fp(,)34 b(then)h Fz(1)27
b Fu(\024)h Fy(d)p Fp(;)586 3254 y Fz(\()p Fy(ii)p Fz(\))42
b(1)28 b Fu(\024)g Fy(e)p Fp(;)35 b(and)553 3370 y Fz(\()p
Fy(iii)p Fz(\))858 3652 y Fy(J)h Fz(=3D)1052 3482 y Fo( )1136
3558 y(M)1131 3770 y Fx(j)t Fw(2)p Fn(K)1257 3779 y Fm(1)1308
3652 y Fy(M)1402 3667 y Fl(1)p Fx(;j)1493 3482 y Fo(!)1589
3558 y(M)1756 3482 y( )1840 3558 y(M)1835 3770 y Fx(j)t
Fw(2)p Fn(K)1961 3779 y Fm(2)2012 3652 y Fy(M)2106 3667
y Fl(2)p Fx(;j)2197 3482 y Fo(!)2293 3558 y(M)2460 3452
y(0)2460 3632 y(@)2553 3558 y(M)2547 3770 y Fx(j)t Fw(2)p
Fn(K)2673 3778 y Fs(p)2725 3652 y Fy(M)2819 3667 y Fx(p;j)2911
3452 y Fo(1)2911 3632 y(A)3015 3652 y Fy(;)770 3929 y
Fp(wher)-5 b(e)803 4090 y Fz(\(1\))41 b Fu(j)p Fk(K)1064
4105 y Fl(1)1103 4090 y Fu(j)21 b Fz(+)h(1)28 b(=3D)f(2\007)22
b(+)g Fy(d)p Fp(,)803 4206 y Fz(\(2\))41 b Fu(j)p Fk(K)1064
4221 y Fl(2)1103 4206 y Fu(j)27 b Fz(=3D)g(1)22 b Fu(\000)h
Fz(\007)p Fp(,)35 b(and)802 4322 y Fz(\()p Fy(p)p Fz(\))42
b Fu(j)p Fk(K)1064 4337 y Fx(p)1103 4322 y Fu(j)22 b
Fz(+)g(1)27 b(=3D)h Fy(e)p Fp(.)456 4545 y(The)34 b(invariants)g
Fy(d)p Fp(,)g Fy(e)p Fp(,)h(and)g Fz(\007)f=
 Fp(determine)g(the)h(mo)-5
b(dule)35 b Fy(J)44 b Fp(uniquely.)555 4807 y Fz(F)-8
b(or)38 b Fy(p)h=
 Fz(=3D)f(2)h(using)h([MS,)h(Theorem)f(3)f(and)g
(Corollary)g(3],)h(along)f(with)g(our)456 4923=
 y(theorem)33
b(ab)s(o)m(v)m(e,)g(w)m(e)h(obtain)f(the)g(next)g(corollary)-8
b(.)456 5099 y FF(Corollary)38 b(2.)j Fp(Now)35 b(let)f
Fy(G)g Fp(b)-5 b(e)34 b(a)g(cyclic)g(gr)-5 b(oup)34 b(of)g(or)-5
b(der)34 b Fz(2)p Fp(.)45 b(Then)33 b(an)h Fv(F)3239
5114 y Fl(2)3278 5099 y Fz([)p Fy(G)p Fz(])p Fp(-)456
5216 y(mo)-5 b(dule)37 b Fy(J)46 b Fp(is)38 b(r)-5=
 b(e)g(alizable)36
b(as)i(an)f Fv(F)1751 5231 y Fl(2)1790 5216 y Fz([Gal\()p
Fy(K)r(=3DF)14 b Fz(\)])p Fp(-mo)-5 b(dule)36 b Fy(K)2742
5179 y Fw(\002)2801 5216 y Fy(=3DK)2940 5179 y Fw(\002)p
Fl(2)3072 5216 y Fp(for)h(some)p eop
%%Page: 3 3
3 2 bop 826 251 a Fq(GALOIS)33=
 b(MODULE)f(CONSTR)n(UCTION)h(AND)f
(CLASSIFICA)-6 b(TION)330 b(3)456 450 y Fp(quadr)-5 b(atic)24
b(extension)g Fy(K)r(=3DF)38 b Fp(with)25 b(its)g(arithmetic)g
(invariants)f Fy(d)p Fz(\()p Fy(K)r(=3DF)14 b Fz(\))p Fp(,)25
b Fy(e)p Fz(\()p Fy(K)r(=3DF)14 b Fz(\))p Fp(,)456 566
y(and)35 b Fz(\007\()p Fy(K)r(=3DF)14 b Fz(\))36 b Fp(c)-5
b(oinciding)34 b(with)j Fy(d)p Fp(,)f Fy(e)p Fp(,)h(and)e
Fz(\007)c Fu(2)f(f)p Fz(0)p Fy(;)17 b Fz(1)p Fu(g)p Fp(,)36
b(r)-5 b(esp)g(e)g(ctively,)36 b(if)g(and)456 683 y(only)e(if)h
Fy(d)p Fp(,)f Fy(e)p Fp(,)h(and)g Fz(\007)f=
 Fp(satisfy)h(the)g(c)-5
b(onditions)34 b(b)-5 b(elow.)679 898 y Fu(\017)41 b
Fp(if)35 b Fz(\007)27 b(=3D)h(0)p Fp(,)34 b(then)h Fz(1)28
b Fu(\024)g Fy(d)p Fp(,)679 1015 y Fu(\017)41 b Fp(if)35
b Fz(\007)27 b(=3D)h(1)p Fp(,)34 b(then)h Fz(1)28 b Fu(\024)g
Fy(e)p Fp(.)555 1231 y(In)34 b(this)h(c)-5 b(ase)1241
1454 y Fy(J)37 b Fz(=3D)1435 1284 y Fo( )1519 1359 y(M)1514
1571 y Fx(j)t Fw(2)p Fn(K)1640 1580 y Fm(1)1691 1454
y Fy(M)1785 1469 y Fl(1)p Fx(;j)1876 1284 y Fo(!)1972
1359 y(M)2139 1284 y( )2223 1359 y(M)2218 1571 y Fx(j)t
Fw(2)p Fn(K)2344 1580 y Fm(2)2395 1454 y Fy(M)2489 1469
y Fl(2)p Fx(;j)2580 1284 y Fo(!)456 1712 y Fp(wher)-5
b(e)604 1928 y Fz(\(1\))41 b Fu(j)p Fk(K)865 1943 y Fl(1)904
1928 y Fu(j)22 b Fz(+)g(1)27 b(=3D)h(2\007)21 b(+)h Fy(d)35
b Fp(and)604 2045 y Fz(\(2\))41 b Fu(j)p Fk(K)865 2060
y Fl(2)904 2045 y Fu(j)22 b Fz(+)g(\007)27 b(=3D)h Fy(e)p
Fp(.)555 2261 y(Mor)-5 b(e)g(over,)34 b(such)f(a)g(mo)-5
b(dule)33 b Fy(J)43 b Fp(is)33 b(determine)-5=
 b(d)32
b(uniquely)i(by)g(the)f(invariants)456 2377 y Fz(2\007)21
b(+)h Fy(d)35 b Fp(and)f(by)h Fy(e)23 b Fu(\000)f Fz(\007)35
b Fp(if)g Fy(e)g Fp(is)f(\014nite)h(and)f(by)h Fy(e)g
Fp(alone)f(if)h Fy(e)g Fp(is)g(in\014nite.)555 2625 y
Fz(If)g Fy(p)d(>)g Fz(2)j(then)h(t)m(w)m(o)g Fv(F)1401
2640 y Fx(p)1441 2625 y Fz([)p Fy(G)p=
 Fz(]-mo)s(dules)f(are)h
(isomorphic)g(if)f(and)g(only)h(if)f(their)456 2742=
 y(in)m(v)-5
b(arian)m(ts)31 b Fy(d)p Fz(,)f Fy(e)p=
 Fz(,)h(and)f(\007)f(are)h(the)h
(same,)g(b)m(y)g([MS)q(,)f(Corollary)g(2].)43 b(Th)m(us)31
b(w)m(e)g(see)456 2858 y(in)i(particular)h(that)f(if)h
Fy(p)29 b(>)g Fz(2,)34 b(then)g(the)g(arithmetic)g(in)m(v)-5
b(arian)m(ts)35 b(of)e Fy(J)42 b Fz(dep)s(end)456 2974
y(only)33 b(up)s(on)f(the)h(isomorphism)h(t)m(yp)s(e)g(of)e(the)h
Fv(F)2214 2989 y Fx(p)2254 2974 y Fz([)p Fy(G)p Fz(]-mo)s(dule)f
Fy(J)9 b Fz(.)555 3170 y(In)45 b(the)h(case)g=
 Fy(p)i
Fz(=3D)h(2,)f(w)m(e)d(see)i(from)d([MS)q(,)k(Corollary)d(2])g(again)f
(that)h(the)456 3286 y(arithmetic)29 b(in)m(v)-5 b(arian)m(ts)30
b Fy(d)p Fz(,)g Fy(e)p Fz(,)g(and)f(\007)f(determine)j(our)e(mo)s(dule)
g Fv(F)2890 3301 y Fl(2)2930 3286 y Fz([)p Fy(G)p=
 Fz(],)g(but)h(t)m(w)m
(o)456 3402 y(isomorphic)d Fv(F)1004 3417 y Fl(2)1043
3402 y Fz([)p Fy(G)p=
 Fz(]-mo)s(dules)g(ma)m(y)f(ha)m(v)m(e)i
(di\013eren)m(t)f(arithmetic)g(in)m(v)-5 b(arian)m(ts;)29
b(see)456 3518 y([MS,)j(Corollary)g(3].)43=
 b(Here)33
b(is)f(a)g(v)m(ery)h(simple,)h(concrete)e(example)i(illustrating)456
3635 y(this)46 b(p)s(ossibilit)m(y)-8 b(.)87 b(Let)46
b Fy(K)1466 3650 y Fl(1)1506 3635 y Fy(=3DF)1618 3650 y
Fl(1)1703 3635 y=
 Fz(b)s(e)g(a)g(quadratic)h(extension)h(of)e(\014nite)h
(\014elds)456 3751 y(of)e(c)m(haracteristic)j(not)e(2,)k
Fy(F)1569 3766 y Fl(2)1659 3751 y Fz(=3D)h Fv(R)p Fz(\(\()p
Fy(t)p Fz(\)\))46=
 b(b)s(e)g(a)g(\014eld)h(of)f(p)s(o)m(w)m(er)h(series)
h(with)456 3870 y(co)s(e\016cien)m(ts)26 b(in)e(real)g(n)m(um)m(b)s
(ers)i Fv(R)p Fz(,)f(and)g Fy(K)2001 3885 y Fl(2)2068
3870 y Fz(=3D)i Fy(F)2234 3885 y Fl(2)2274 3870 y Fz(\()2312
3792 y Fu(p)p 2395 3792 127 4 v 78 x(\000)p Fz(1\).)40
b(Then)26 b(b)s(oth)d(mo)s(dules)456 3988 y Fy(K)546
3946 y Fw(\002)539 4012 y Fl(1)605 3988 y Fy(=3DK)744 3946
y Fw(\002)p Fl(2)737 4012 y(1)861 3988 y Fz(and)g Fy(K)1131
3946 y Fw(\002)1124 4012 y Fl(2)1191 3988 y Fy(=3DK)1330
3946 y Fw(\002)p Fl(2)1323 4012 y(2)1447 3988 y=
 Fz(are)g(isomorphic)i
(to)d(a)h(trivial)h Fv(F)2608 4003 y Fl(2)2647 3988 y
Fz([)p Fy(G)p Fz(]-mo)s(dule)g Fv(F)3207 4003 y Fl(2)3246
3988 y Fz(,)h(but)456 4104 y(their)31=
 b(arithmetic)h(in)m(v)-5
b(arian)m(ts)32 b(\()p Fy(d)1692 4119 y Fx(i)1720 4104
y Fy(;)17 b(e)1809 4119 y Fx(i)1837 4104 y Fy(;)g Fz(\007)1957
4119 y Fx(i)1985 4104 y Fz(\))31 b(are)g(\(0)p Fy(;)17
b Fz(1)p Fy(;)g Fz(1\))29 b(for)i Fy(i)c Fz(=3D)h(1)j(and)g(\(2)p
Fy(;)17 b Fz(0)p Fy(;)g Fz(0\))456 4220 y(for)32 b Fy(i)c
Fz(=3D)f(2.)1289 4497 y(1.)48 b Fj(Not)-7 b(a)g(tion)40
b(and)e(Stra)-7 b(tegy)555 4751 y Fz(In)29=
 b(all)g(that)g(follo)m(ws)h
Fy(F)42 b Fz(denotes)30 b(a)f(\014eld,)h Fy(F)2174 4714
y Fw(\002)2261 4751 y Fz(=3D)d Fy(F)i Fu(n)15 b(f)p Fz(0)p
Fu(g)27 b Fz(the)j(m)m(ultiplicativ)m(e)456 4867 y(group)39
b(of)g Fy(F)14 b Fz(,)40 b Fy(p)g Fz(a)f(prime)h(n)m(um)m(b)s(er,)i
(and)e Fy(F)2125 4831 y Fw(\002)2184 4867 y Fy(=3DF)2310
4831 y Fw(\002)p Fx(p)2443 4867 y Fz(the)g(group)f(of)g
Fy(p)p Fz(th-p)s(o)m(w)m(er)456 4983 y(classes)47 b(of)f
Fy(F)14 b Fz(.)83 b(F)-8 b(or)45 b(eac)m(h)i Fy(f)61
b Fu(2)50 b Fy(F)1814 4947 y Fw(\002)1919 4983 y=
 Fz(w)m(e)d(denote)f(b)
m(y)h([)p Fy(f)11 b Fz(])46 b(the)g(class)h(of)f Fy(f)56
b Fz(in)456 5099 y Fy(F)533 5063 y Fw(\002)591 5099 y
Fy(=3DF)717 5063 y Fw(\002)p Fx(p)811 5099 y Fz(.)48 b(F)-8
b(or)33 b(eac)m(h)i(subset)g Fy(A)f Fz(of)g Fy(F)1880
5063 y Fw(\002)1972 5099 y Fz(w)m(e)h(denote)g(b)m(y)g([)p
Fy(A)p Fz(])f(the)g(set)h(of)e(classes)456 5216 y Fu(f)p
Fz([)p Fy(a)p Fz(])f Fu(j)h Fy(a)27 b Fu(2)h Fy(A)p Fu(g)33
b Fz(and)g(b)m(y)g Fu(h)p Fz([)p Fy(A)p Fz(])p=
 Fu(i)f
Fz(the)h(subgroup)h(of)e Fy(F)2374 5179 y Fw(\002)2433
5216 y Fy(=3DF)2559 5179 y Fw(\002)p Fx(p)2685 5216 y=
 Fz(generated)h(b)m
(y)h([)p Fy(A)p Fz(].)p eop
%%Page: 4 4
4 3 bop 456 255 a Fq(4)797 b(J)1340 236 y(\023)1330 255
y(AN)26 b(MIN)1638 236 y(\023)1628 255 y(A)1695 236 y(\024)1686
255 y(C)f(AND)f(JOHN)h(SW)-9 b(ALLO)n(W)555 450 y=
 Fz(W)h(e)29
b(denote)h(b)m(y)g Fy(\030)1205 465 y Fx(p)1273 450 y
Fz(a)e(primitiv)m(e)j Fy(p)p Fz(th)d(ro)s(ot)g(of)h(unit)m(y)g(in)g
Fy(F)14 b Fz(.)42 b(\(Some)30 b(\014elds)g(will)456=
 566
y(b)s(e)g(assumed)i(to)e(con)m(tain)h(suc)m(h)g(a)f(primitiv)m(e)i
Fy(p)p Fz(th)f(ro)s(ot;)f(for)g(the)g(other)h(\014elds)g(in)456
683=
 y(this)k(pap)s(er,)g(w)m(e)h(will)f(pro)m(v)m(e)h(that)e(a)g
(primitiv)m(e)i Fy(p)p Fz(th)f(ro)s(ot)e(is)j(con)m(tained)f(in)g(the)
456 799 y(\014eld.\))58 b(Observ)m(e)39=
 b(that)e(our)g(assumption)i
(that)e(there)h(exists)h(a)e(primitiv)m(e)i Fy(p)p Fz(th)456
915 y(ro)s(ot)31 b(of)h(unit)m(y)i(implies)g(that)f(c)m(har\()p
Fy(F)14 b Fz(\))27 b Fu(6)p Fz(=3D)h Fy(p)p Fz(.)555 1118
y(F)-8 b(or)43 b(a)f(Galois)h(extension)i Fy(K)r(=3DF)14
b Fz(,)45 b(Gal\()p Fy(K)r(=3DF)14 b Fz(\))42=
 b(denotes)i(the)g(Galois)f
(group)456 1234 y(and)34 b Fy(N)725 1249 y Fx(K)q(=3DF)913
1234 y Fz(denotes)h(the)g(norm)f(map)g(from)g Fy(K)41
b Fz(to)34 b Fy(F)14 b Fz(.)47 b(W)-8 b(e)35 b(denote)g(b)m(y)f
Fy(F)3238 1198 y Fx(s)3309 1234 y Fz(the)456 1350=
 y(separable)j
(closure)g(of)e Fy(F)50 b Fz(and)35 b Fy(G)1713 1365
y Fx(F)1808 1350 y Fz(the)h(absolute)h(Galois)e(group)h(Gal)o(\()p
Fy(F)3217 1314 y Fx(s)3254 1350 y Fy(=3DF)14 b Fz(\).)456
1466 y(As)43 b(usual,)j Fy(H)994 1430 y Fx(i)1022 1466
y Fz(\()p Fy(G)1137 1481 y Fx(F)1195 1466 y Fy(;)17 b
Fv(F)1300 1481 y Fx(p)1339 1466 y Fz(\))43=
 b(are)g(Galois)f(cohomology)
h(groups)g(of)f Fy(F)57 b Fz(with)43 b(co)s(ef-)456=
 1582
y(\014cien)m(ts)g(in)e Fv(F)976 1597 y Fx(p)1016 1582
y Fz(.)69 b(Since)43=
 b(all)e(absolute)h(Galois)f(groups)h(will)g(b)s(e)
f(pro-)p Fy(p)p Fz(-groups,)456 1699 y(all)c(considered)j
Fv(F)1140 1714 y Fx(p)1217 1699 y=
 Fz(mo)s(dules)f(are)f(trivial.)59
b(Finally)-8 b(,)40 b(let)e Fu(j)p Fy(B)5 b Fu(j)37=
 b
Fz(b)s(e)h(the)g(cardinal)456 1815 y(n)m(um)m(b)s(er)c(of)e(a)g(set)h
Fy(B)5 b Fz(.)555 2017 y(First)26=
 b(observ)m(e)i(that)d(the)i
(conditions)f(on)g Fy(d)p Fz(,)h Fy(e)p=
 Fz(,)g(and)f(\007)g(listed)h
(in)f(our)g(theorem)456 2134 y(ab)s(o)m(v)m(e)33 b(are)g(necessary:)604
2360 y(\(1\))41 b(If)33 b(\007)27 b(=3D)h(0,)k(then)h Fy(\030)1448
2375 y Fx(p)1527 2360 y Fy(=3D)-61 b Fu(2)28 b Fy(N)1687
2376 y Fx(K)q(=3DF)1842 2360 y Fz(\()p Fy(K)1970 2324 y
Fw(\002)2029 2360 y Fz(\))k(and)h(hence)1306 2531 y Fy(d)28
b Fz(=3D)f(dim)1651 2546 y Ft(F)1692 2554 y Fs(p)1748 2531
y Fy(F)1825 2489 y Fw(\002)1884 2531 y Fy(=3D)-5 b(N)2006
2546 y Fx(K)q(=3DF)2160 2531 y Fz(\()p Fy(K)2288 2489 y
Fw(\002)2347 2531 y Fz(\))28 b Fu(\025)g Fz(1;)604 2700
y(\(2\))41 b(If)g Fy(p)h(>)f Fz(2)g(and)g Fy(K)48 b Fz(=3D)42
b Fy(F)14 b Fz(\()1763 2667 y Fs(p)1740 2628 y Fu(p)p
1823 2628 52 4 v 72 x Fy(a)p Fz(\))41 b(for)f(a)h(suitable)h
Fy(a)f Fu(2)i Fy(F)2850 2664 y Fw(\002)2936 2700 y Fu(n)28
b Fy(F)3091 2664 y Fw(\002)p Fx(p)3185 2700 y Fz(,)43
b(then)770 2816 y Fy(a)28 b Fz(=3D)g Fy(N)1031 2832 y Fx(K)q(=3DF)1185
2816 y Fz(\()1250 2783 y Fs(p)1227 2745 y Fu(p)p 1310
2745 V 71 x Fy(a)p Fz(\))33 b(and)g(hence)1400 2987 y
Fy(e)27 b Fz(=3D)h(dim)1739 3002 y Ft(F)1780 3010 y Fs(p)1836
2987 y Fy(N)1914 3002 y Fx(K)q(=3DF)2068 2987 y Fz(\()p
Fy(K)2196 2945 y Fw(\002)2255 2987 y Fz(\))g Fy(>)g Fz(0)p
Fy(:)604 3156 y Fz(\(3\))41 b(If)j Fy(p)k Fz(=3D)g(2,)f(\007)g(=3D)h(1,)f
(and)d Fy(K)55 b Fz(=3D)48 b Fy(F)14 b Fz(\()2171 3084
y Fu(p)p 2253 3084 V 2253 3156 a Fy(a)p Fz(\))45=
 b(for)e(a)h(suitable)i
Fy(a)i Fu(2)g Fy(F)3306 3120 y Fw(\002)3395 3156 y Fu(n)770
3272 y Fy(F)847 3236 y Fw(\002)p Fl(2)941 3272 y Fz(,)d(then)f
Fu(\000)p Fz(1)h Fu(2)g Fy(N)1606 3288 y Fx(K)q(=3DF)1760
3272 y Fz(\()p Fy(K)1888 3236 y Fw(\002)1947 3272 y=
 Fz(\))e(since)h
(\007)g(=3D)h(1.)73 b(Consequen)m(tly)46 b Fy(a)f Fu(2)770
3389 y Fy(N)848 3405 y Fx(K)q(=3DF)1002 3389 y Fz(\()p
Fy(K)1130 3353 y Fw(\002)1190 3389 y Fz(\),)32 b(and)h(th)m(us)g(1)28
b Fu(\024)g Fy(e)p Fz(.)456 3616 y(Therefore)38=
 b(in)f(order)g(to)f
(pro)m(v)m(e)j(Theorem)f(1)e(when)j Fy(p)c(>)f=
 Fz(2,)k(it)f(is)h
(su\016cien)m(t)h(to)456 3732 y(sho)m(w,)30=
 b(for)d(eac)m(h)i(cardinal)
g(n)m(um)m(b)s(ers)g Fy(d)p Fz(,)g Fy(e)f=
 Fz(as)g(ab)s(o)m(v)m(e,)i
(for)e(eac)m(h)h(\007)e Fu(2)h(f)p Fz(0)p Fy(;)17 b Fz(1)p
Fu(g)p Fz(,)28 b(and)456 3848 y(for)k(eac)m(h)h(prime)h(n)m(um)m(b)s
(er)f Fy(p)28 b(>)g Fz(2,)k(the)h(existence)i(of)d(a)h(\014eld)g
Fy(F)46 b Fz(suc)m(h)34 b(that:)679 4075 y Fu(\017)41
b Fy(F)46 b Fz(con)m(tains)34 b(a)e(primitiv)m(e)j Fy(p)p
Fz(th)d(ro)s(ot)g Fy(\030)2186 4090 y Fx(p)2225 4075
y Fz(;)679 4191 y Fu(\017)41 b Fy(F)847 4155 y Fw(\002)906
4191 y Fy(=3DF)1032 4155 y Fw(\002)p Fx(p)1158 4191 y=
 Fz(decomp)s(oses)35
b(in)m(tro)e(a)f(direct)h(sum)h(of)e(subgroups)1399 4356
y Fy(F)1476 4315 y Fw(\002)1535 4356 y Fy(=3DF)1661 4315
y Fw(\002)p Fx(p)1782 4356 y Fz(=3D)c Fy(D)d Fu(\010)d(h)p
Fz([)p Fy(a)p Fz(])p Fu(i)g(\010)h Fy(E)6 b(;)770 4521
y Fz(where)34 b(dim)1215 4536 y Ft(F)1256 4544 y Fs(p)1295
4521 y Fz(\()p Fu(h)p Fz([)p Fy(a)p Fz(])p Fu(i)22 b(\010)h
Fy(E)6 b Fz(\))28 b(=3D)f Fy(e)33 b Fz(and,)g(setting)g
Fy(K)i Fz(=3D)27 b Fy(F)14 b Fz(\()2864 4487 y Fs(p)2840
4449 y Fu(p)p 2923 4449 V 72 x Fy(a)q Fz(\),)803 4638
y(\(1\))41 b([)p Fy(N)1074 4654 y Fx(K)q(=3DF)1228 4638
y Fz(\()p Fy(K)1356 4602 y Fw(\002)1415 4638 y Fz(\)])28
b(=3D)g Fu(h)p Fz([)p Fy(a)p Fz(])p Fu(i)22 b(\010)g Fy(E)6
b Fz(;)803 4755 y(\(2\))41 b(dim)1131 4770 y Ft(F)1172
4778 y Fs(p)1212 4755 y Fz(\()p Fy(F)1327 4719 y Fw(\002)1386
4755 y Fy(=3D)-5 b(N)1508 4771 y Fx(K)q(=3DF)1662 4755 y
Fz(\()p Fy(K)1790 4719 y Fw(\002)1849 4755 y Fz(\)\))27
b(=3D)h(dim)2218 4770 y Ft(F)2259 4778 y Fs(p)2316 4755
y Fy(D)i Fz(=3D)e Fy(d)p Fz(;)k(and)803 4873 y(\(3\))41
b(\007)27 b(=3D)h(0)k(if)h(and)f(only)h(if)g Fy(\030)1883
4888 y Fx(p)1961 4873 y Fy(=3D)-60 b Fu(2)28 b Fy(N)2122
4888 y Fx(K)q(=3DF)2276 4873 y Fz(\()p Fy(K)2404 4836 y
Fw(\002)2463 4873 y Fz(\).)555 5099 y(In)37 b(the)g(case)g
Fy(p)d=
 Fz(=3D)g(2)i(and)h(\007)d(=3D)g(1)i(w)m(e)h(use)g(the)g(same)g
(conditions)h(as)e(ab)s(o)m(v)m(e,)456 5216 y(and)45
b(if)f Fy(p)49 b Fz(=3D)f(2)d(and)g(\007)j(=3D)h(0)44=
 b(w)m(e)i(require)g
(instead)g(that)f Fy(e)j Fz(=3D)h(dim)3068 5231 y Ft(F)3109
5240 y Fm(2)3164 5216 y Fy(E)i Fz(and)p eop
%%Page: 5 5
5 4 bop 826 251 a Fq(GALOIS)33=
 b(MODULE)f(CONSTR)n(UCTION)h(AND)f
(CLASSIFICA)-6 b(TION)330 b(5)456 450 y Fy(d)35 b Fz(=3D)h(dim)816
465 y Ft(F)857 474 y Fm(2)896 450 y Fz(\()p Fy(D)28 b
Fu(\010)e(h)p Fz([)p Fy(a)p Fz(])p Fu(i)p Fz(\).)57=
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b(latter)e(condition)h(is)g(imp)s(osed)g(b)s(ecause)h
Fu(\000)p Fz(1)48 b Fy(=3D)-61 b Fu(2)456 566 y Fy(N)534
586 y Fx(F)10 b Fl(\()616 538 y Fw(p)p 674 538 38 3 v
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Fz(\()986 495 y Fu(p)p 1069 495 52 4 v 71 x Fy(a)p Fz(\))1158
530 y Fw(\002)1217 566 y Fz(\))32 b(if)h(and)g(only)g(if)f
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Fl(\()2203 538 y Fw(p)p 2262 538 38 3 v 48 x Fx(a)p Fl(\))p
Fx(=3DF)2421 566 y Fz(\()p Fy(F)k Fz(\()2574 495 y Fu(p)p
2656 495 52 4 v 2656 566 a Fy(a)p Fz(\))2745 530 y Fw(\002)2804
566 y Fz(\).)555 769 y(Our)23=
 b(strategy)h(is)g(to)f(in)m(terpret)h
(the)g(required)g(conditions)h(on)e Fy(F)2899 733 y Fw(\002)2957
769 y Fy(=3DF)3083 733 y Fw(\002)p Fx(p)3201 769 y Fz(ab)s(o)m(v)m(e)456
886 y(in)32 b(terms)h(of)f(Galois)g(cohomology)-8 b(.)43
b(W)-8 b(e)33 b(then)g(observ)m(e)h(that)e(these)h(conditions)456
1002 y(are)39 b(satis\014ed)i(if)e Fy(G)1180 1017 y Fx(F)1278
1002 y Fz(is)h(a)f(free)h(pro)s(duct,)i(in)d(the)h(category)g(of)f
(pro-)p Fy(p)p Fz(-groups,)456 1118 y(of)f(suitable)j(pro-)p
Fy(p)p Fz(-groups)d Fy(G)1597 1133 y Fl(1)1676 1118 y
Fz(and)h Fy(G)1949 1133 y Fl(2)1989 1118 y=
 Fz(,)i(and)e(\014nally)h(w)m
(e)h(use)f(the)g(v)m(ery)g(nice)456 1234=
 y(theorem)32
b(pro)m(v)m(ed)h(b)m(y)g(Efrat)f(and)g(Haran)f(whic)m(h)i(guaran)m
(tees)g(the)f(existence)j(of)456 1351 y(a)28 b(\014eld)i(with)g
Fy(G)1037 1366 y Fx(F)1124 1351 y Fz(ab)s(o)m(v)m(e.)43
b(This)30=
 b(is)g(one)f(of)g(the)g(k)m(ey)i(results)f(used)g(in)f(our)g
(pap)s(er.)456 1524 y FF(Theorem)42 b(2.)i Fp(\(Efr)-5
b(at-Har)g(an;)40 b(se)-5 b(e)38 b=
 Fz([EH)q(,)f(Prop)s(osition)g(1.3])p
Fp(\))h(L)-5 b(et)39 b Fy(F)3076 1539 y Fl(1)3116 1524
y Fy(;)17 b(:)g(:)g(:)f(;)h(F)3398 1539 y Fx(n)456 1640
y Fp(b)-5 b(e)26 b(\014elds)g(of)h(e)-5 b(qual)26 b(char)-5
b(acteristic)27 b(such)f(that)i Fy(G)2233 1655 y Fx(F)2278
1664 y Fm(1)2316 1640 y Fy(;)17 b(:)g(:)g(:)f(;)h(G)2612
1655 y Fx(F)2657 1663 y Fs(n)2730 1640 y Fp(ar)-5 b(e)26
b(pr)-5 b(o-)p Fy(p)p Fp(-gr)g(oups.)456 1756 y(Then)34
b(ther)-5 b(e)34 b(exists)h(a)g(\014eld)f Fy(F)48 b=
 Fp(of)35
b(the)g(same)f(char)-5 b(acteristic)34 b(such)h(that)1487
1913 y Fy(G)1564 1928 y Fx(F)1650 1886 y Fu(\030)1651
1918 y Fz(=3D)1755 1913 y Fy(G)1832 1928 y Fx(F)1877 1937
y Fm(1)1938 1913 y Fy(?)22 b Fu(\001)17 b(\001)g(\001)j
Fy(?)i(G)2295 1928 y Fx(F)2340 1936 y Fs(n)2386 1913
y Fy(;)456 2070 y Fp(wher)-5 b(e)34 b(the)h(pr)-5 b(o)g(duct)35
b(is)f(fr)-5 b(e)g(e)35 b(in)f(the)h(c)-5 b(ate)g(gory)35
b(of)g(pr)-5 b(o-)p Fy(p)p Fp(-gr)g(oups.)456 2326 y
Fz(In)35=
 b(order)f(to)h(apply)g(this)h(theorem,)g(w)m(e)f(sho)m(w)h
(the)f(existence)j(of)c(the)h(\014elds)h Fy(F)3405 2341
y Fl(1)456 2442 y Fz(and)27 b Fy(F)703 2457 y Fl(2)769
2442 y Fz(suc)m(h)i(that)e Fy(G)1267 2457 y Fx(F)1312
2466 y Fm(1)1377 2442 y Fz(and)h Fy(G)1639 2457 y Fx(F)1684
2466 y Fm(2)1749 2442 y Fz(are)f(prescrib)s(ed)i(Galois)e(groups)g
Fy(G)3050 2457 y Fl(1)3117 2442 y Fz(and)g Fy(G)3378
2457 y Fl(2)3417 2442 y Fz(.)456 2559 y(W)-8 b(e)33=
 b(use)g(the)g(tec)m
(hniques)i(of)d(henselian)j(v)-5 b(aluations)32=
 b(and)h(formal)f(p)s(o)
m(w)m(er)i(series)456 2675 y(to)e(construct)i(\014elds)f
Fy(F)1316 2690 y Fl(1)1388 2675 y Fz(and)g Fy(F)1641
2690 y Fl(2)1681 2675 y Fz(.)1698 2955 y(2.)49 b Fj(Lemmas)456
3211 y Fz(2.1.)f FF(V)-9 b(alued)41 b(\014elds)g Fy(F)1387
3226 y Fl(1)1466 3211 y=
 FF(with)f(prescrib)s(ed)h(residue)f(\014eld)h
Fy(F)2970 3226 y Fl(0)3050 3211 y FF(and)f(v)-6 b(al-)456
3328 y(uation)38 b(group)f Fz(\000)p FF(.)555 3526 y
Fz(Let)48 b Fy(v)j Fz(b)s(e)d(a)f(v)-5 b(aluation)47
b(on)g(a)h(\014eld)g Fy(F)2063 3541 y Fl(1)2102 3526
y Fz(,)k(written)c(additiv)m(ely)-8 b(.)90 b(Then)48
b(w)m(e)456 3642 y(denote)28 b(b)m(y)g Fy(A)968 3657
y Fx(v)1037 3642 y Fz(the)g(v)-5 b(aluation)27 b(ring)g
Fu(f)p Fy(f)39 b Fu(2)28 b Fy(F)2109 3657 y Fl(1)2176
3642 y Fu(j)f Fy(v)t Fz(\()p Fy(f)11 b Fz(\))27 b Fu(\025)h
Fz(0)p Fu(g)p Fz(;)h(b)m(y)f Fy(M)2928 3657 y Fx(v)2997
3642 y Fz(the)g(unique)456 3759 y(maximal)d(ideal)h Fu(f)p
Fy(f)38 b Fu(2)28 b Fy(A)1375 3774 y Fx(v)1441 3759 y
Fu(j)d Fy(v)t Fz(\()p Fy(f)11 b Fz(\))27 b Fy(>)h Fz(0)p
Fu(g)c Fz(of)h Fy(A)2111 3774 y Fx(v)2152 3759 y Fz(;)j(b)m(y)e
Fy(F)2398 3774 y Fx(v)2464 3759 y Fz(the)f(residue)i(\014eld)f
Fy(A)3225 3774 y Fx(v)3266 3759 y Fy(=3D)-5 b(M)3404 3774
y Fx(v)456 3875 y Fz(of)28 b Fy(v)t=
 Fz(;)i(b)m(y)f(\000)g(the)g(v)-5
b(aluation)29 b(group)g Fy(v)t Fz(\()p Fy(F)1917 3834
y Fw(\002)1903 3899 y Fl(1)1975 3875 y Fz(\))g(of)f=
 Fy(v)t
Fz(;)i(and)f(b)m(y)g Fy(U)40 b Fz(the)29 b(group)g Fy(A)3190
3890 y Fx(v)3245 3875 y Fu(n)14 b Fy(M)3403 3890 y Fx(v)456
3991 y Fz(of)32 b(units)h(of)f Fy(v)t Fz(.)555 4190=
 y(The)e(follo)m
(wing)e(lemma)i(is)f(w)m(ell)g(kno)m(wn)h(and)f(w)m(e)g(shall)g(omit)g
(its)g(straigh)m(tfor-)456 4306 y(w)m(ard)k(pro)s(of.)456
4479 y FF(Lemma)43 b(1.)g Fp(L)-5 b(et)39 b Fy(F)1222
4494 y Fl(1)1300 4479 y Fp(b)-5 b(e)38 b(a)g(value)-5
b(d)38 b(\014eld)f(with)h(valuation)g Fy(v)t=
 Fp(,)h(valuation)f(gr)-5
b(oup)456 4596 y Fz(\000)31 b(=3D)g Fy(v)t Fz(\()p Fy(F)821
4554 y Fw(\002)807 4620 y Fl(1)879 4596 y Fz(\))p Fp(,)37
b(and)g(gr)-5 b(oup)36 b(of)h(units)g Fy(U)10 b Fp(.)51
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Fz(=3D)h(c)m(har\()p Fy(F)3122 4611 y Fl(1)3162 4596 y
Fz(\))k Fp(ther)-5 b(e)456 4712 y(exists)34 b(an)h(isomorphism)1289
4869 y Fy(')27 b Fz(:)h Fy(F)1512 4828 y Fw(\002)1498
4893 y Fl(1)1571 4869 y Fy(=3DF)1697 4822 y Fw(\002)p Fx(p)1683
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4828 y Fx(p)2265 4869 y Fu(\010)23 b Fz(\000)p Fy(=3Dp)p
Fz(\000)p Fy(:)456 5026 y Fp(In)34 b(p)-5 b(articular)1015
5183 y Fz(dim)1177 5198 y Ft(F)1218 5206 y Fs(p)1275
5183 y Fy(F)1352 5141 y Fw(\002)1338 5207 y Fl(1)1411
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5183 y Fz(=3D)28 b(dim)1924 5198 y Ft(F)1965 5206 y Fs(p)2022
5183 y Fy(U)5 b(=3DU)2218 5141 y Fx(p)2280 5183 y Fz(+)22
b(dim)2541 5198 y Ft(F)2582 5206 y Fs(p)2638 5183 y Fz(\000)p
Fy(=3Dp)p Fz(\000)p Fy(:)p eop
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6 5 bop 456 255 a Fq(6)797 b(J)1340 236 y(\023)1330 255
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255 y(C)f(AND)f(JOHN)h(SW)-9 b(ALLO)n(W)555 450 y=
 Fz(It)38
b(is)g(w)m(ell-kno)m(wn)i(that)e(for)f(eac)m(h)i(\014eld)g
Fy(F)2157 465 y Fl(0)2234 450 y=
 Fz(and)f(for)f(eac)m(h)i(totally)f
(ordered)456 566 y(ab)s(elian)45=
 b(group)h(\000,)j(there)d(exists)i(a)d
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566 y Fu(!)456 683 y Fz(\000)29 b Fu([)g(f1g)42 b=
 Fz(suc)m(h)i(that)f
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683 y Fz(is)g(isomorphic)f(to)g Fy(F)3027 698 y Fl(0)3109
683 y Fz(and)g(the)456 799 y(v)-5 b(aluation)32 b(group)g(is)i(\000.)
555 1004 y(In)f(order)g(to)f(construct)i(suc)m(h)g(a)e(\014eld,)i(set)
757 1175 y Fy(F)820 1190 y Fl(1)887 1175 y Fz(=3D)27 b
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y Fl(0)2006 1175 y Fu(j)49 b Fz(supp)q(\()p Fy(f)11 b
Fz(\))32 b(is)h(w)m(ell-ordered)r Fu(g)p Fy(:)456 1346
y Fz(Th)m(us)42 b(a)e(t)m(ypical)i(elemen)m(t)g Fy(f)52
b Fu(2)41 b Fy(F)1766 1361 y Fl(1)1846 1346 y=
 Fz(can)g(b)s(e)g(written)
g(as)g(a)f(formal)g(sum)i Fy(f)52 b Fz(=3D)456 1388 y Fo(P)561
1491 y Fx(g)r Fw(2)p Fl(\000)709 1462 y Fy(a)760 1477
y Fx(g)800 1462 y Fy(t)835 1426 y Fx(g)916 1462 y=
 Fz(suc)m(h)42
b(that)f(the)h(set)f(supp)q(\()p Fy(f)11 b Fz(\))42 b(:=3D)g
Fu(f)p Fy(g)j Fu(2)d Fy(G)f Fu(j)f Fy(a)2711 1477 y Fx(g)2793
1462 y Fu(6)p Fz(=3D)i(0)p Fu(g)f Fz(is)g(a)g(w)m(ell-)456
1587 y(ordered)29 b(subset)h(of)f Fy(G)p Fz(.)42 b(The)30
b(v)-5 b(aluation)28 b Fy(v)33 b Fz(on)28 b Fy(f)40 b
Fz(is)29 b(de\014ned)h(as:)42 b Fy(v)t Fz(\(0\))27 b(=3D)h
Fu(1)g Fz(and)456 1704 y Fy(v)t Fz(\()p Fy(f)11 b Fz(\))36
b(=3D)h(min)18 b(supp)q(\()p Fy(f)11 b Fz(\))37 b(for)h
Fy(f)48 b Fu(6)p Fz(=3D)37 b(0.)61 b(An)38=
 b(imp)s(ortan)m(t)g(prop)s
(ert)m(y)h(of)f(the)h(v)-5 b(alued)456 1820 y(\014eld)31
b Fy(F)728 1835 y Fl(1)798 1820 y=
 Fz(as)f(ab)s(o)m(v)m(e)h(is)g(the)g
(fact)f(that)g(it)g(is)h(henselian.)44 b(\(See)32=
 b(for)d(example)j
([Rib,)456 1936 y(\(1.3\)].\))40 b(In)24=
 b(what)g(follo)m(ws)h(w)m(e)g
(will)g(iden)m(tify)g Fy(F)2169 1951 y Fx(v)2234 1936
y Fz(with)g Fy(F)2511 1951 y Fl(0)2550 1936 y Fz(.)41
b(W)-8 b(e)24 b(will)h(also)f(assume)456 2052 y(that)32
b(c)m(har)h Fy(F)944 2067 y Fl(0)1011 2052 y Fu(6)p Fz(=3D)28
b Fy(p)p Fz(.)555 2257 y(W)-8 b(e)25=
 b(will)g(b)s(e)f(particularly)h
(in)m(terested)i(in)d(con)m(trolling)h(the)g Fy(p)p=
 Fz(th-p)s(o)m(w)m
(er)f(classes)456 2374 y(of)e(suc)m(h)j(a)d(\014eld.)41
b(T)-8 b(o)24=
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b(These)35 b(groups)e(will)g(b)s(e)g(direct)g(sums)h(of)592
2687 y Fv(Z)658 2702 y Fl(\()p Fx(p)p Fl(\))781 2687
y Fz(:=3D)912 2576 y Fo(n)988 2620 y Fy(a)p 988 2664 52
4 v 993 2755 a(b)1077 2687 y Fu(2)28 b Fv(Q)1281 2602
y Fo(\014)1281 2662 y(\014)1347 2687 y Fy(a;)17 b(b)28
b Fu(2)g Fv(Z)p Fy(;)17 b(b)28 b Fu(6)p Fz(=3D)g(0;)49
b(if)32 b Fy(a)c Fu(6)p Fz(=3D)g(0)k(then)h(\()p Fy(a;)17
b(b)p Fz(\))28 b(=3D)g(1)p Fy(;)17 b(p)27 b Fv(-)g Fy(b)3197
2576 y Fo(o)3280 2687 y Fy(:)456 2888 y Fz(Observ)m(e)33
b(that)e Fv(Z)1105 2903 y Fl(\()p Fx(p)p Fl(\))1232 2888
y Fz(is)h(the)g(v)-5 b(aluation)31 b(ring)g(of)g(a)h
Fy(p)p Fz(-adic)f(v)-5 b(aluation)31 b(on)h Fv(Q)p Fz(.)43
b(Let)456 3004 y Fy(I)d Fz(b)s(e)33 b(an)m(y)g(non-empt)m(y)-8
b(,)34 b(w)m(ell-ordered)g(set.)44 b(Then)34 b(set)986
3201 y(\000)28 b(=3D)f Fv(Z)1244 3150 y Fl(\()p Fx(I)5
b Fl(\))1244 3232 y(\()p Fx(p)p Fl(\))1367 3201 y Fz(:=3D)1498
3090 y Fo(n)1564 3201 y Fy(\015)33 b Fz(:)28 b Fy(I)35
b Fu(!)28 b Fv(Z)1975 3217 y Fl(\()p Fx(p)p Fl(\))2102
3086 y Fo(\014)2102 3146 y(\014)2102 3206 y(\014)2168
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b Fy(<)g(\015)842 3767 y Fl(2)881 3752 y Fz(\()p=
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Fz(\))k(for)f(the)i(least)f(elemen)m(t)i Fy(i)30 b Fu(2)h
Fy(I)41 b Fz(suc)m(h)36 b(that)d Fy(\015)2664 3767 y
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4306 y(b)s(elo)m(w\))48=
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4422 y(follo)m(wing)36 b(lemma.)57 b(This)37=
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7 6 bop 826 251 a Fq(GALOIS)33=
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1509 y(c)m(haracter)40 b(mapping)f Fy(G)1372 1524 y Fx(F)1417
1533 y Fm(0)1494 1509 y=
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 b(particular,)g(in)g
(order)f(that)g(our)456 2128 y(cyclic)46 b(extension)h
Fy(K)55 b Fz(=3D)49 b Fy(F)14 b Fz(\()1585 2094 y Fs(p)1562
2056 y Fu(p)p 1645 2056 52 4 v 72 x Fy(a)p Fz(\))45=
 b(ha)m(v)m(e)h(the)
f(desired)h(in)m(v)-5 b(arian)m(t)46 b(\007)i(=3D)h(0,)e(w)m(e)456
2244 y(require)37 b(that)f Fy(K)43 b Fz(do)s(es)37=
 b(not)f(em)m(b)s(ed)
h(in)g(a)e(cyclic)j(Galois)e(extension)i Fy(L)e Fz(o)m(v)m(er)i
Fy(F)456 2360 y Fz(with)f(degree)h([)p Fy(L)e Fz(:)f
Fy(F)14 b Fz(])36 b(=3D)f Fy(p)1481 2324 y Fl(2)1520 2360
y Fz(,)j(for)f(this)h(is)f(equiv)-5 b(alen)m(t)39 b(to)e
Fy(\030)2671 2375 y Fx(p)2757 2360 y Fy(=3D)-61 b Fu(2)36
b Fy(N)2925 2375 y Fx(K)q(=3DF)3079 2360 y Fz(\()p Fy(K)3207
2324 y Fw(\002)3266 2360 y Fz(\))h(b)m(y)456 2476=
 y([A,)32
b(Theorem)i(3].)555 2680 y(T)-8 b(o)47=
 b(ensure)i(that)d(this)i(nonem)m
(b)s(eddabilit)m(y)i(condition)d(holds,)52 b(as)47 b(w)m(ell)h(as)456
2797 y(to)37=
 b(ensure)j(that)d(a)h(certain)h(nonab)s(elian)f(group)g
(of)f(order)h Fy(p)2732 2760 y Fl(3)2810 2797 y=
 Fz(do)s(es)g(not)g(o)s
(ccur)456 2913=
 y(as)g(a)f(Galois)h(group)g(o)m(v)m(er)g(the)h(\014eld,)
h(w)m(e)f(c)m(ho)s(ose)f(residue)i(\014elds)f Fy(F)3004
2928 y Fl(0)3081 2913 y Fz(with)g(ab-)456 3029=
 y(solute)j(Galois)g
(groups)g(taking)g(a)g(sp)s(ecial)g(form,)i(and)e(it)g(is)h(also)f(con)
m(v)m(enien)m(t)456 3145 y(to)h(require)j(that)e Fu(j)p
Fy(F)1254 3104 y Fw(\002)1240 3170 y Fl(0)1312 3145 y
Fy(=3DF)1438 3098 y Fw(\002)p Fx(p)1424 3170 y Fl(0)1532
3145 y Fu(j)g Fz(is)h(small.)79 b(As)45=
 b(it)f(turns)h(out,)i(w)m(e)e
(ma)m(y)g(c)m(ho)s(ose)456 3261 y(some)29=
 b(suitable)g(algebraic)g
(in\014nite)h(extension)g(of)e Fv(Q)p Fz(.)42=
 b(Finitely)30
b(generated)f(pro-)456 3378 y Fy(p)p=
 Fz(-absolute)39
b(Galois)f(groups)i(o)m(v)m(er)g Fv(Q)f=
 Fz(and)g(more)g(generally)h(an)
m(y)g(global)e(\014eld,)456 3494 y(w)m(ere)31=
 b(nicely)g(classi\014ed)h
(in)e([E2)q(].)42 b(\(See)31=
 b(also)f([E1])g(and)g([JP)q(])g(for)f
(related)i(results)456 3610 y(and)h(tec)m(hniques.\))555
3814 y(The)i(extensions)h(w)m(e)e(will)g(need)h(for)e
Fy(p)c(>)f Fz(2)33 b(are)f(giv)m(en)i(in)f(the)g(follo)m(wing)456
3995 y FF(Lemma)c(5.)36 b Fz([E2,)26 b(page)f(84])i Fp(F)-7
b(or)26 b(e)-5 b(ach)27 b(prime)g Fy(p)g(>)h Fz(2)f Fp(ther)-5
b(e)27 b(exists)h(an)f(algebr)-5 b(aic)456 4112 y(extension)33
b Fy(F)949 4127 y Fl(0)p Fx(;p)1079 4112 y Fp(of)i=
 Fv(Q)g
Fp(such)g(that)1243 4284 y Fy(G)1320 4299 y Fx(F)1365
4308 y Fm(0)p Fs(;p)1482 4284 y Fz(=3D)28 b Fu(h)p Fy(\033)n(;)17
b(\034)46 b Fu(j)35 b Fy(\033)t(\034)11 b(\033)2044 4243
y Fw(\000)p Fl(1)2166 4284 y Fz(=3D)28 b Fy(\034)2323 4243
y Fx(p)p Fl(+1)2453 4284 y Fu(i)2492 4299 y Fh(pr)l(o-)p
Fx(p)456 4453 y Fp(wher)-5 b(e)34 b(the)h(pr)-5=
 b(esentation)34
b(is)h(in)f(the)h(c)-5 b(ate)g(gory)35 b(of)f(pr)-5 b(o-)p
Fy(p)p Fp(-gr)g(oups.)555 4722 y Fz(Observ)m(e)31=
 b(that)d(the)h
(maximal)g(ab)s(elian)g(extension)h Fy(F)2530 4686 y
Fx(ab)2516 4746 y Fl(0)p Fx(;p)2639 4722 y Fz(of)e Fy(F)2809
4737 y Fl(0)p Fx(;p)2933 4722 y Fz(has)g Fy(G)3179 4686
y Fx(ab)3179 4747 y(F)3224 4756 y Fm(0)p Fs(;p)3341 4722
y Fz(:=3D)456 4856 y(Gal)o(\()p Fy(F)723 4820 y Fx(ab)709
4881 y Fl(0)p Fx(;p)804 4856 y Fy(=3DF)916 4871 y Fl(0)p
Fx(;p)1010 4856 y Fz(\))33 b(equal)g(to)1183 5042 y Fy(G)1260
5001 y Fx(ab)1260 5067 y(F)1305 5076 y Fm(0)p Fs(;p)1422
5014 y Fu(\030)1423 5046 y Fz(=3D)1527 5042 y Fu(h)5 b
Fz(\026)-54 b Fy(\033)t(;)22 b Fz(\026)-54 b Fy(\034)44
b Fu(j)37 b Fz(\026)-54 b Fy(\034)1868 5001 y Fx(p)1936
5042 y Fz(=3D)27 b(1)p Fu(i)h Fz(=3D)f Fv(Z)2324 5057 y Fx(p)2386
5042 y Fu(\002)c Fv(Z)p Fy(=3Dp)p Fv(Z)456 5216 y Fz(for)32
b(all)g(primes)i Fy(p)28 b(>)f Fz(2.)p eop
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9 8 bop 826 251 a Fq(GALOIS)33=
 b(MODULE)f(CONSTR)n(UCTION)h(AND)f
(CLASSIFICA)-6 b(TION)330 b(9)456 450 y Fz(2.4.)48 b
FF(Field)39 b(arithmetic)e(and)i(free)e(pro-)p Fy(p)h
FF(pro)s(ducts.)555 650 y Fz(In)32=
 b(this)g(section)g(w)m(e)g(collect)g
(lemmas)h(giving)e(information)g(ab)s(out)g(a)g(\014eld)h
Fy(F)456 766 y Fz(deriv)m(ed)40 b(from)e(the)h(structure)h(of)e
Fy(G)1831 781 y Fx(F)1890 766 y Fz(,)i(esp)s(ecially)h(when)e
Fy(G)2740 781 y Fx(F)2838 766 y Fz(is)g(a)f(free)h(pro-)p
Fy(p)456 882 y Fz(pro)s(duct)32 b(of)h(t)m(w)m(o)g(pro-)p
Fy(p)f Fz(groups)h Fy(G)1764 897 y Fx(F)1809 906 y Fm(1)1879
882 y Fz(and)g Fy(G)2146 897 y Fx(F)2191 906 y Fm(2)2229
882 y Fz(.)555 1081 y(First)25=
 b(w)m(e)h(record)g(a)e(lemma)i
(detecting)g(the)g(presence)h(of)d(primitiv)m(e)j=
 Fy(p)p
Fz(th)e(ro)s(ots)456 1198 y(of)32 b(unit)m(y)h(in)g(a)g(\014eld)g
Fy(F)14 b Fz(,)32 b(based)i(only)f(on)f(the)h(structure)h(of)e
Fy(G)2759 1213 y Fx(F)2818 1198 y Fz(.)456 1372 y FF(Lemma)39
b(6.)j Fp(Supp)-5 b(ose)34 b(that)i Fy(p)28 b(>)g Fz(2)34
b Fp(and)h(that)g Fy(F)49 b Fp(is)35=
 b(a)g(\014eld)f(with)h
Fz(c)m(har)q(\()p Fy(F)14 b Fz(\))27 b Fu(6)p Fz(=3D)h
Fy(p)456 1488 y Fp(and)34 b Fy(G)722 1503 y Fx(F)815
1488 y Fp(pr)-5 b(o-)p Fy(p)p Fp(.)45 b(Then)34 b Fy(\030)1407
1503 y Fx(p)1474 1488 y Fu(2)28 b Fy(F)1645 1452 y Fw(\002)1703
1488 y Fp(.)456 1738 y(Pr)-5 b(o)g(of.)41 b Fz(Because)23
b(c)m(har)q(\()p Fy(F)14 b Fz(\))27 b Fu(6)p Fz(=3D)h=
 Fy(p)p
Fz(,)c(there)e(exists)i(a)e(primitiv)m(e)h Fy(p)p=
 Fz(th)f(ro)s(ot)g
Fy(\030)3061 1753 y Fx(p)3122 1738 y Fz(of)f(unit)m(y)456
1854 y(in)34 b Fy(F)648 1818 y Fx(s)684 1854 y Fz(.)48
b(If)34 b Fy(\030)901 1869 y Fx(p)970 1854 y Fu(2)d Fy(F)1144
1818 y Fx(s)1203 1854 y Fu(n)23 b Fy(F)48 b Fz(then)34
b Fy(F)14 b Fz(\()p Fy(\030)1768 1869 y Fx(p)1807 1854
y Fz(\))p Fy(=3DF)47 b Fz(is)35=
 b(a)f(non)m(trivial)g(Galois)g(extension)
i(of)456 1970 y(degree)i([)p Fy(F)14 b Fz(\()p Fy(\030)949
1985 y Fx(p)988 1970 y Fz(\))35 b(:)g Fy(F)14 b Fz(])35
b Fy(<)h(p)p Fz(.)56 b(Therefore)39 b Fy(G)2030 1985
y Fx(F)2125 1970 y Fz(has)f(a)e(non)m(trivial)i(\014nite)g(quotien)m(t)
456 2087 y(of)29 b(order)i(coprime)g(with)f Fy(p)p Fz(.)43
b(This)31 b(con)m(tradicts)h(our)e(assumption)h(that)f
Fy(G)3211 2102 y Fx(F)3300 2087 y Fz(is)h(a)456 2203
y(pro-)p Fy(p)p Fz(-group.)42 b(Hence)34 b Fy(\030)1358
2218 y Fx(p)1425 2203 y Fu(2)28 b Fy(F)1596 2167 y Fw(\002)1687
2203 y Fz(as)33 b(asserted.)1192 b Fi(\003)555 2453 y
Fz(No)m(w)34 b(supp)s(ose)g(that)f Fy(G)1434 2468 y Fx(F)1521
2453 y Fz(=3D)28 b Fy(G)1702 2468 y Fx(F)1747 2477 y Fm(1)1808
2453 y Fy(?)22 b(G)1956 2468 y Fx(F)2001 2477 y Fm(2)2072
2453 y Fz(for)32 b(pro-)p Fy(p)h Fz(absolute)g(Galois)g(groups)456
2569 y Fy(G)533 2584 y Fx(F)591 2569 y Fz(,)k Fy(G)732
2584 y Fx(F)777 2593 y Fm(1)815 2569 y Fz(,)g(and)f Fy(G)1149
2584 y Fx(F)1194 2593 y Fm(2)1232 2569 y=
 Fz(,)h(where)g(the)f(free)g
(pro)s(duct)g(is)h(tak)m(en)f(in)g(the)h(category)f(of)456
2685 y(pro-)p Fy(p)p Fz(-groups.)61 b(F)-8 b(rom)38=
 b([N,)j(\(4.3\))d
(Satz])h(w)m(e)h(see)f(that)g(the)g(restriction)h(homo-)456
2801 y(morphism)930 2959 y(res)28 b(:)g Fy(H)1221 2917
y Fl(1)1260 2959 y Fz(\()p Fy(G)1375 2974 y Fx(F)1433
2959 y Fy(;)17 b Fv(F)1538 2974 y Fx(p)1578 2959 y Fz(\))27
b Fu(\000)-16 b(!)28 b Fy(H)1921 2917 y Fl(1)1959 2959
y Fz(\()p Fy(G)2074 2974 y Fx(F)2119 2983 y Fm(1)2158
2959 y Fy(;)17 b Fv(F)2263 2974 y Fx(p)2302 2959 y Fz(\))22
b Fu(\010)h Fy(H)2551 2917 y Fl(1)2590 2959 y Fz(\()p
Fy(G)2705 2974 y Fx(F)2750 2983 y Fm(2)2788 2959 y Fy(;)17
b Fv(F)2893 2974 y Fx(p)2932 2959 y Fz(\))350 b(\(1\))456
3116 y(is)24 b(an)g(isomorphism.)42 b(No)m(w)25 b(giv)m(en)g(\()p
Fy(f)11 b Fz(\))1872 3131 y Fx(F)1954 3116 y Fz(in)24
b Fy(H)2148 3080 y Fl(1)2187 3116 y Fz(\()p Fy(G)2302
3131 y Fx(F)2360 3116 y Fy(;)17 b Fv(F)2465 3131 y Fx(p)2505
3116 y Fz(\),)25 b(w)m(e)g(denote)g(the)f(image)456 3232
y(res\()p Fy(f)11 b Fz(\))710 3247 y Fx(F)801 3232 y
Fz(b)m(y)1382 3349 y(res)q(\()p Fy(f)g Fz(\))1637 3364
y Fx(F)1723 3349 y Fz(=3D)28 b(\()p Fy(f)11 b Fz(\))1962
3364 y Fx(G)2017 3375 y Fs(F)2056 3390 y Fm(1)2120 3349
y Fu(\010)22 b Fz(\()p Fy(f)11 b Fz(\))2354 3364 y Fx(G)2409
3375 y Fs(F)2448 3390 y Fm(2)2491 3349 y Fy(:)456 3493
y Fz(This)27 b(notation)f(distinguishes,)31=
 b(then,)d(b)s(et)m(w)m(een)
h(\()p Fy(f)11 b Fz(\))2405 3508 y Fx(F)2450 3517 y Fm(1)2488
3493 y Fz(,)28 b(whic)m(h)g(denotes)f Fy(')3226 3508
y Fx(F)3271 3517 y Fm(1)3310 3493 y Fz(\()p Fy(f)11 b
Fz(\))456 3610 y(for)27 b Fy(f)38 b Fu(2)29 b Fy(F)858
3568 y Fw(\002)844 3634 y Fl(1)916 3610 y Fz(,)g(and)f(\()p
Fy(f)11 b Fz(\))1292 3625 y Fx(G)1347 3636 y Fs(F)1386
3651 y Fm(1)1428 3610 y Fz(,)29=
 b(whic)m(h)h(denotes)f(the)g(pro)5
b(jection)28 b(of)g(res)17 b Fy(')3036 3625 y Fx(F)3095
3610 y Fz(\()p Fy(f)11 b Fz(\))27 b(on)m(to)456 3728
y(the)33 b(\014rst)g(summand.)555 3928=
 y(One)g(w)m(a)m(y)h(of)e(in)m
(terpreting)i(this)f(restriction)h(map)f(is)g(with)g(the)g(follo)m
(wing)456 4102 y FF(Lemma)c(7.)36 b Fp(L)-5 b(et)27 b
Fy(G)1203 4117 y Fx(F)1290 4102 y Fz(=3D)g Fy(G)1470 4117
y Fx(F)1515 4126 y Fm(1)1559 4102 y Fy(?)5 b(G)1690 4117
y Fx(F)1735 4126 y Fm(2)1801 4102 y Fp(b)-5 b(e)27 b(pr)-5
b(o-)p Fy(p)27 b Fp(absolute)g(Galois)g(gr)-5 b(oups)27
b(of)g(\014elds)456 4218 y(c)-5 b(ontaining)35 b(a)g
Fy(p)p Fp(th)i(r)-5 b(o)g(ot)36 b(of)g(unity,)g(and)g(supp)-5
b(ose)35 b(that)i(we)e(have)h(the)g(fol)5 b(lowing)456
4335 y(se)-5 b(quenc)g(e:)1416 4476 y Fy(G)1493 4491
y Fx(F)1616 4420 y Fl(can)1751 4449 y Fg(/)p Ff(/)29
b Fg(/)p Ff(/)p 1580 4451 200 4 v 1809 4471 a Fy(G)1886
4486 y Fx(F)1931 4495 y Fm(1)2168 4449 y Fg(/)p Ff(/)g
Fg(/)p Ff(/)p 1998 4451 V 2226 4474 a Fv(Z)p Fy(=3Dp)p
Fv(Z)p Fy(:)456 4626 y Fp(Her)-5 b(e)33 b(the)g(c)-5
b(anonic)g(al)32 b(map)h Fz(can)h Fp(is)f(an)g(identity)g(on)g
Fy(G)2477 4641 y Fx(F)2522 4650 y Fm(1)2594 4626 y=
 Fp(and)g(c)-5
b(ontains)32 b Fy(G)3243 4641 y Fx(F)3288 4650 y Fm(2)3360
4626 y Fp(in)456 4743 y(its)j(kernel.)555 4942=
 y(Then)44
b(the)g(right-hand)g(surje)-5 b(ction)44=
 b(and)f(the)i(c)-5
b(omp)g(ose)g(d)43 b(surje)-5 b(ction)44 b(c)-5 b(orr)g(e-)456
5058 y(sp)g(ond)44 b(to)i(\014elds)e Fy(K)1210 5073 y
Fl(1)1297 5058 y Fz(=3D)j Fy(F)1483 5073 y Fl(1)1523 5058
y Fz(\()1588 5034 y Fs(p)1565 4994 y Fu(p)p 1648 4994
91 4 v 64 x Fy(a)1699 5073 y Fl(1)1739 5058 y Fz(\))e
Fp(and)g Fy(K)54 b Fz(=3D)47 b Fy(F)14 b Fz(\()2425 5025
y Fs(p)2401 4987 y Fu(p)p 2484 4987 52 4 v 71 x Fy(a)q
Fz(\))p Fp(,)48 b(r)-5 b(esp)g(e)g(ctively,)47 b(wher)-5
b(e)456 5175 y Fz(\()p Fy(a)p Fz(\))583 5190 y Fx(G)638
5201 y Fs(F)677 5216 y Fm(1)747 5175 y Fz(=3D)27 b(\()p
Fy(a)939 5190 y Fl(1)979 5175 y Fz(\))1017 5190 y Fx(F)1062
5199 y Fm(1)1135 5175 y Fp(and)34 b Fz(\()p Fy(a)p Fz(\))1451
5190 y Fx(G)1506 5201 y Fs(F)1545 5216 y Fm(2)1616 5175
y Fz(=3D)27 b(0)p Fp(.)p eop
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10 9 bop 456 255 a Fq(10)759 b(J)1340 236 y(\023)1330
255 y(AN)26 b(MIN)1638 236 y(\023)1628 255 y(A)1695 236
y(\024)1686 255 y(C)f(AND)f(JOHN)h(SW)-9 b(ALLO)n(W)456
450 y Fp(Pr)k(o)g(of.)41 b Fz(The)27=
 b(surjections)g(are)f(con)m(tin)m
(uous)i(homomorphisms,)h(hence)e(elemen)m(ts)456 566
y(of)c Fy(H)647 530 y Fl(1)686 566 y Fz(\()p Fy(G)801
581 y Fx(F)846 590 y Fm(1)885 566 y Fy(;)17 b Fv(F)990
581 y Fx(p)1029 566 y Fz(\))24 b(and)g Fy(H)1361 530
y Fl(1)1400 566 y Fz(\()p Fy(G)1515 581 y Fx(F)1574 566
y Fy(;)17 b Fv(F)1679 581 y Fx(p)1718 566 y=
 Fz(\),)26
b(resp)s(ectiv)m(ely)-8 b(,)29 b(and)24 b(the)h(righ)m(t-hand)f
(surjec-)456 683=
 y(tion)e(is)g(clearly)i(the)e(restriction)h(of)f(the)h
(comp)s(osed)g(surjection.)41 b(The)23 b(remainder)456
799 y(follo)m(ws)33 b(b)m(y)g(Kummer)h(theory)-8 b(.)1757
b Fi(\003)456 980 y FF(Lemma)45 b(8.)f Fp(L)-5 b(et)40
b Fy(G)1240 995 y Fx(F)1335 980 y Fz(=3D)c Fy(G)1524 995
y Fx(F)1569 1004 y Fm(1)1633 980 y Fy(?)25 b(G)1784 995
y Fx(F)1829 1004 y Fm(2)1907 980 y Fp(b)-5 b(e)39 b(pr)-5
b(o-)p Fy(p)39 b Fp(absolute)h(Galois)f(gr)-5 b(oups)39
b(and)456 1097 y(supp)-5 b(ose)34 b(that)h Fz(\()p Fy(a)p
Fz(\))1138 1112 y Fx(F)1232 1097 y Fp(and)f Fz(\()p Fy(b)p
Fz(\))1538 1112 y Fx(F)1632 1097 y Fp(satisfy)h Fz(\()p
Fy(a)p Fz(\))2063 1112 y Fx(G)2118 1123 y Fs(F)2157 1138
y Fm(1)2227 1097 y Fz(=3D)27 b(\()p Fy(b)p Fz(\))2447 1112
y Fx(G)2502 1123 y Fs(F)2541 1138 y Fm(2)2612 1097 y
Fz(=3D)g(0)p Fp(.)45 b(Then)1308 1283 y Fz(\()p Fy(a)p
Fz(\))1435 1298 y Fx(F)1515 1283 y Fu([)23 b Fz(\()p
Fy(b)p Fz(\))1721 1298 y Fx(F)1808 1283 y Fz(=3D)k(0)h
Fu(2)g Fy(H)2171 1242 y Fl(2)2210 1283 y Fz(\()p Fy(G)2325
1298 y Fx(F)2383 1283 y Fy(;)17 b Fv(F)2488 1298 y Fx(p)2527
1283 y Fz(\))p Fy(:)555 1553 y Fz(The)45=
 b(lemma)f(follo)m(ws)g(from)f
([N,)j(\(4.1\))d(Satz])h(and)f(from)g([Ris)q(,)j(Prop.)e(7.3,)456
1669 y(page)31 b(191].)43 b(Ho)m(w)m(ev)m(er,)34=
 b(w)m(e)f(pro)m(v)m(e)
g(our)f(lemma)g(b)m(y)h(translating)f(the)g(cup)g(pro)s(d-)456
1786=
 y(ucts)d(in)m(to)f(obstructions)h(to)f(basic)h(Galois)f(em)m(b)s
(edding)i(problems,)g(yielding)f(an)456 1902=
 y(in)m(teresting)34
b(Galois-theoretic)f(v)-5 b(arian)m(t)32 b(of)g(the)h(pro)s(of.)456
2172 y Fp(Pr)-5 b(o)g(of.)41 b Fz(If)d(\()p Fy(a)p=
 Fz(\))g(=3D)g(0)g(or)g
(\()p Fy(b)p Fz(\))h(=3D)e(0,)j(w)m(e)g(are)f(done.)62
b(Otherwise,)41 b(the)e(conditions)456 2288 y(\()p Fy(a)p
Fz(\))583 2303 y Fx(G)638 2314 y Fs(F)677 2329 y Fm(1)747
2288 y Fz(=3D)27 b(\()p Fy(b)p Fz(\))967 2303 y Fx(G)1022
2314 y Fs(F)1061 2329 y Fm(2)1132 2288 y Fz(=3D)g(0)32
b(imply)i(that)e(\()p Fy(a)p Fz(\))g(and)h(\()p=
 Fy(b)p
Fz(\))f(are)g(linearly)i(indep)s(enden)m(t)g(in)456 2416
y Fy(H)545 2380 y Fl(1)584 2416 y Fz(\()p Fy(G)699 2431
y Fx(F)757 2416 y Fy(;)17 b Fv(F)862 2431 y Fx(p)901
2416 y Fz(\).)555 2621 y(No)m(w)33 b(let)g Fy(H)999 2637
y Fx(p)1035 2619 y Fm(3)1106 2621 y=
 Fz(b)s(e)g(the)g(Heisen)m(b)s(erg)h
(group)e(of)h(order)f Fy(p)2594 2585 y Fl(3)2634 2621
y Fz(:)897 2790 y Fy(H)978 2807 y Fx(p)1014 2788 y Fm(3)1079
2790 y Fz(=3D)c Fu(h)p Fy(v)1269 2805 y Fl(1)1308 2790
y Fy(;)17 b(v)1399 2805 y Fl(2)1438 2790 y Fy(;)g(w)35
b Fu(j)d Fy(v)1698 2743 y Fx(p)1694 2815 y Fl(1)1765
2790 y Fz(=3D)c Fy(v)1920 2743 y Fx(p)1916 2815 y Fl(2)1987
2790 y Fz(=3D)g Fy(w)2164 2749 y Fx(p)2230 2790 y Fz(=3D)g(1)p
Fy(;)17 b(v)2474 2805 y Fl(2)2513 2790 y Fy(v)2560 2805
y Fl(1)2627 2790 y Fz(=3D)28 b Fy(w)s(v)2851 2805 y Fl(1)2890
2790 y Fy(v)2937 2805 y Fl(2)2976 2790 y Fy(;)1222 2941
y Fz([)p Fy(v)1296 2956 y Fl(1)1335 2941 y Fy(;)17 b(w)s
Fz(])27 b(=3D)h([)p Fy(v)1684 2956 y Fl(2)1723 2941 y Fy(;)17
b(w)s Fz(])27 b(=3D)h(1)p Fu(i)456 3111 y Fz(In)33 b(the)g(case)g
Fy(p)28 b Fz(=3D)f(2,)33 b Fy(H)1322 3126 y Fl(8)1393 3111
y Fz(is)g(the)g(familiar)g(dihedral)g(group)g Fy(D)2757
3126 y Fl(4)2796 3111 y Fz(.)555 3316 y(By)41 b([M)q(,)h(Corollary)-8
b(,)43 b(page)d(523)g(and)h(Theorem)h(3\(A\)],)e(if)g(\()p
Fy(a)p Fz(\))h(and)g(\()p Fy(b)p Fz(\))f(are)456 3432
y(linearly)31 b(indep)s(enden)m(t,)i(then)e(\()p Fy(a)p
Fz(\))18 b Fu([)g Fz(\()p Fy(b)p Fz(\))28 b(=3D)f(0)j(if)g(and)h(only)g
(if)f Fy(H)2812 3449 y Fx(p)2848 3430 y Fm(3)2916 3432
y Fz(is)h(the)g(Galois)456 3563 y(group)i(Gal)o(\()p
Fy(M)5 b(=3DF)14 b Fz(\))33 b(of)g(a)g(Galois)g(extension)j
Fy(M)44 b Fz(of)33 b Fy(F)47 b Fz(con)m(taining)34 b
Fy(F)14 b Fz(\()3123 3529 y Fs(p)3100 3491 y Fu(p)p 3183
3491 52 4 v 72 x Fy(a;)3305 3514 y Fs(p)3282 3478 y Fu(p)p
3365 3478 42 4 v 85 x Fy(b)p Fz(\))456 3679 y(in)32=
 b(suc)m(h)i(a)f(w)m
(a)m(y)h(that)570 3858 y Fy(H)651 3875 y Fx(p)687 3856
y Fm(3)725 3858 y Fy(=3D)p Fu(h)p Fy(v)860 3873 y Fl(1)899
3858 y Fy(;)17 b(w)s Fu(i)27 b Fz(=3D)g(Gal\()p Fy(F)14
b Fz(\()1518 3819 y Fs(p)1495 3781 y Fu(p)p 1578 3781
52 4 v 77 x Fy(a)p Fz(\))p Fy(=3DF)g Fz(\))32 b(and)g Fy(H)2133
3875 y Fx(p)2169 3856 y Fm(3)2208 3858 y Fy(=3D)p Fu(h)p
Fy(v)2343 3873 y Fl(2)2382 3858 y Fy(;)17 b(w)s Fu(i)26
b Fz(=3D)i(Gal)o(\()p Fy(F)14 b Fz(\()3000 3803 y Fs(p)2977
3768 y Fu(p)p 3060 3768 42 4 v 90 x Fy(b)q Fz(\))p Fy(=3DF)g
Fz(\))p Fy(:)555 4116 y Fz(No)m(w)33=
 b(consider)h(the)f(comm)m(utativ)m
(e)i(diagram)1882 4294 y Fy(G)1959 4309 y Fx(F)1658 4402
y(\016)1689 4411 y Fm(1)1648 4519 y Fg(|)p Ff(|)1625
4537 y Fg(|)p Ff(|)1625 4538 y Fe(z)1655 4513 y(z)1685
4488 y(z)1715 4464 y(z)1744 4439 y(z)1774 4414 y(z)1804
4389 y(z)1833 4365 y(z)1979 4633 y Fx(\014)1950 4859
y Fg(\017)p Ff(\017)1950 4888 y Fg(\017)p Ff(\017)p 1948
4890 4 4 v 1948 4873 V 1948 4857 V 1948 4841 V 1948 4825
V 1948 4809 V 1948 4793 V 1948 4776 V 1948 4760 V 1948
4744 V 1948 4728 V 1948 4712 V 1948 4696 V 1948 4679
V 1948 4663 V 1948 4647 V 1948 4631 V 1948 4615 V 1948
4599 V 1948 4582 V 1948 4566 V 1948 4550 V 1948 4534
V 1948 4518 V 1948 4502 V 1948 4485 V 1948 4469 V 1948
4453 V 1948 4437 V 1948 4421 V 1948 4404 V 1948 4388
V 1948 4372 V 1948 4356 V 1948 4340 V 2176 4402 a Fx(\016)2207
4411 y Fm(2)2253 4519 y Fg(")p Ff(")2275 4537 y Fg(")p
Ff(")2243 4511 y Fe(D)2213 4486 y(D)2184 4461 y(D)2154
4437 y(D)2124 4412 y(D)2094 4387 y(D)2065 4363 y(D)2035
4338 y(D)1455 4634 y Fy(G)1532 4649 y Fx(F)1577 4658
y Fm(1)1215 4751 y Fx(\013)1260 4760 y Fm(1)1206 4869
y Fg({)p Ff({)1183 4887 y Fg({)p Ff({)p Fe(x)1215 4862
y(x)1247 4837 y(x)1279 4812 y(x)1310 4788 y(x)1342 4763
y(x)1374 4738 y(x)1406 4713 y(x)2285 4634 y Fy(G)2362
4649 y Fx(F)2407 4658 y Fm(2)2605 4751 y Fx(\013)2650
4760 y Fm(2)2694 4869 y Fg(#)p Ff(#)2717 4887 y Fg(#)p
Ff(#)2684 4861 y Fe(F)2652 4837 y(F)2621 4812 y(F)2589
4787 y(F)2557 4762 y(F)2525 4737 y(F)2493 4712 y(F)2461
4688 y(F)967 4991 y Fv(Z)p Fy(=3Dp)p Fv(Z)1206 4945 y Fd(\037)1226
4924 y(\177)1429 4927 y Fl(1)p Fw(7!)p Fx(v)1569 4936
y Fm(1)1843 4966 y Fg(/)p Ff(/)p 1226 4967 617 4 v 1872
4985 a Fy(H)1953 5002 y Fx(p)1989 4983 y Fm(3)2120 5203
y Fw(h)p Fx(v)2181 5212 y Fm(1)2216 5203 y Fx(;w)r Fw(i7!)p
Fl(0;)c Fx(v)2507 5212 y Fm(2)2542 5203 y Fw(7!)p Fl(1)2712
5045 y Fg(8)p Ff(8)p 2053 5046 4 4 v 2055 5047 V 2056
5048 V 2058 5048 V 2060 5049 V 2061 5050 V 2063 5051
V 2064 5052 V 2066 5053 V 2068 5054 V 2069 5054 V 2071
5055 V 2072 5056 V 2074 5057 V 2076 5058 V 2077 5058
V 2079 5059 V 2081 5060 V 2082 5061 V 2084 5062 V 2085
5062 V 2087 5063 V 2089 5064 V 2090 5065 V 2092 5066
V 2094 5066 V 2095 5067 V 2097 5068 V 2099 5069 V 2100
5069 V 2102 5070 V 2104 5071 V 2106 5071 V 2107 5072
V 2109 5073 V 2111 5074 V 2112 5074 V 2114 5075 V 2116
5076 V 2117 5076 V 2119 5077 V 2121 5078 V 2123 5078
V 2124 5079 V 2126 5080 V 2128 5081 V 2129 5081 V 2131
5082 V 2133 5082 V 2135 5083 V 2136 5084 V 2138 5084
V 2140 5085 V 2142 5086 V 2143 5086 V 2145 5087 V 2147
5088 V 2149 5088 V 2150 5089 V 2152 5089 V 2154 5090
V 2156 5090 V 2157 5091 V 2159 5092 V 2161 5092 V 2163
5093 V 2165 5093 V 2166 5094 V 2168 5094 V 2170 5095
V 2172 5096 V 2174 5096 V 2175 5097 V 2177 5097 V 2179
5098 V 2181 5098 V 2183 5099 V 2184 5099 V 2186 5100
V 2188 5100 V 2190 5101 V 2192 5101 V 2193 5102 V 2195
5102 V 2197 5103 V 2199 5103 V 2201 5104 V 2203 5104
V 2204 5105 V 2206 5105 V 2208 5105 V 2210 5106 V 2212
5106 V 2214 5107 V 2215 5107 V 2217 5108 V 2219 5108
V 2221 5108 V 2223 5109 V 2225 5109 V 2226 5110 V 2228
5110 V 2230 5110 V 2232 5111 V 2234 5111 V 2236 5112
V 2238 5112 V 2240 5112 V 2241 5113 V 2243 5113 V 2245
5113 V 2247 5114 V 2249 5114 V 2251 5114 V 2253 5115
V 2254 5115 V 2256 5115 V 2258 5116 V 2260 5116 V 2262
5116 V 2264 5117 V 2266 5117 V 2268 5117 V 2270 5117
V 2271 5118 V 2273 5118 V 2275 5118 V 2277 5119 V 2279
5119 V 2281 5119 V 2283 5119 V 2285 5120 V 2287 5120
V 2288 5120 V 2290 5120 V 2292 5121 V 2294 5121 V 2296
5121 V 2298 5121 V 2300 5121 V 2302 5122 V 2304 5122
V 2306 5122 V 2307 5122 V 2309 5122 V 2311 5123 V 2313
5123 V 2315 5123 V 2317 5123 V 2319 5123 V 2321 5123
V 2323 5124 V 2325 5124 V 2327 5124 V 2329 5124 V 2330
5124 V 2332 5124 V 2334 5124 V 2336 5125 V 2338 5125
V 2340 5125 V 2342 5125 V 2344 5125 V 2346 5125 V 2348
5125 V 2350 5125 V 2352 5125 V 2353 5125 V 2355 5125
V 2357 5125 V 2359 5126 V 2361 5126 V 2363 5126 V 2365
5126 V 2367 5126 V 2369 5126 V 2371 5126 V 2373 5126
V 2375 5126 V 2377 5126 V 2378 5126 V 2380 5126 V 2382
5126 V 2384 5126 V 2386 5126 V 2388 5126 V 2390 5126
V 2392 5126 V 2394 5126 V 2396 5126 V 2398 5126 V 2400
5126 V 2402 5126 V 2403 5126 V 2405 5126 V 2407 5125
V 2409 5125 V 2411 5125 V 2413 5125 V 2415 5125 V 2417
5125 V 2419 5125 V 2421 5125 V 2423 5125 V 2425 5125
V 2426 5125 V 2428 5125 V 2430 5124 V 2432 5124 V 2434
5124 V 2436 5124 V 2438 5124 V 2440 5124 V 2442 5124
V 2444 5123 V 2446 5123 V 2448 5123 V 2449 5123 V 2451
5123 V 2453 5123 V 2455 5122 V 2457 5122 V 2459 5122
V 2461 5122 V 2463 5122 V 2465 5121 V 2467 5121 V 2469
5121 V 2470 5121 V 2472 5121 V 2474 5120 V 2476 5120
V 2478 5120 V 2480 5120 V 2482 5119 V 2484 5119 V 2486
5119 V 2488 5119 V 2489 5118 V 2491 5118 V 2493 5118
V 2495 5117 V 2497 5117 V 2499 5117 V 2501 5117 V 2503
5116 V 2504 5116 V 2506 5116 V 2508 5115 V 2510 5115
V 2512 5115 V 2514 5114 V 2516 5114 V 2518 5114 V 2519
5113 V 2521 5113 V 2523 5113 V 2525 5112 V 2527 5112
V 2529 5112 V 2531 5111 V 2533 5111 V 2534 5110 V 2536
5110 V 2538 5110 V 2540 5109 V 2542 5109 V 2544 5108
V 2546 5108 V 2547 5108 V 2549 5107 V 2551 5107 V 2553
5106 V 2555 5106 V 2557 5105 V 2558 5105 V 2560 5105
V 2562 5104 V 2564 5104 V 2566 5103 V 2568 5103 V 2569
5102 V 2571 5102 V 2573 5101 V 2575 5101 V 2577 5100
V 2578 5100 V 2580 5099 V 2582 5099 V 2584 5098 V 2586
5098 V 2587 5097 V 2589 5097 V 2591 5096 V 2593 5096
V 2595 5095 V 2596 5094 V 2598 5094 V 2600 5093 V 2602
5093 V 2604 5092 V 2605 5092 V 2607 5091 V 2609 5090
V 2611 5090 V 2612 5089 V 2614 5089 V 2616 5088 V 2618
5088 V 2619 5087 V 2621 5086 V 2623 5086 V 2625 5085
V 2626 5084 V 2628 5084 V 2630 5083 V 2632 5082 V 2633
5082 V 2635 5081 V 2637 5081 V 2639 5080 V 2640 5079
V 2642 5078 V 2644 5078 V 2645 5077 V 2647 5076 V 2649
5076 V 2651 5075 V 2652 5074 V 2654 5074 V 2656 5073
V 2657 5072 V 2659 5071 V 2661 5071 V 2662 5070 V 2664
5069 V 2666 5069 V 2667 5068 V 2669 5067 V 2671 5066
V 2672 5066 V 2674 5065 V 2676 5064 V 2677 5063 V 2679
5062 V 2681 5062 V 2682 5061 V 2684 5060 V 2686 5059
V 2687 5058 V 2689 5058 V 2691 5057 V 2692 5056 V 2694
5055 V 2695 5054 V 2697 5054 V 2699 5053 V 2700 5052
V 2702 5051 V 2703 5050 V 2705 5049 V 2707 5048 V 2708
5048 V 2710 5047 V 1252 5203 a Fw(h)p Fx(v)1313 5212
y Fm(2)1348 5203 y Fx(;w)r Fw(i7!)p Fl(0;)h Fx(v)1640
5212 y Fm(1)1675 5203 y Fw(7!)p Fl(1)1188 5045 y Fg(f)p
Ff(f)p 1843 5046 V 1842 5047 V 1840 5048 V 1839 5048
V 1837 5049 V 1836 5050 V 1834 5051 V 1832 5052 V 1831
5053 V 1829 5054 V 1828 5054 V 1826 5055 V 1824 5056
V 1823 5057 V 1821 5058 V 1819 5058 V 1818 5059 V 1816
5060 V 1814 5061 V 1813 5062 V 1811 5062 V 1810 5063
V 1808 5064 V 1806 5065 V 1805 5066 V 1803 5066 V 1801
5067 V 1800 5068 V 1798 5069 V 1796 5069 V 1795 5070
V 1793 5071 V 1791 5071 V 1789 5072 V 1788 5073 V 1786
5074 V 1784 5074 V 1783 5075 V 1781 5076 V 1779 5076
V 1778 5077 V 1776 5078 V 1774 5078 V 1772 5079 V 1771
5080 V 1769 5081 V 1767 5081 V 1765 5082 V 1764 5082
V 1762 5083 V 1760 5084 V 1759 5084 V 1757 5085 V 1755
5086 V 1753 5086 V 1752 5087 V 1750 5088 V 1748 5088
V 1746 5089 V 1744 5089 V 1743 5090 V 1741 5090 V 1739
5091 V 1737 5092 V 1736 5092 V 1734 5093 V 1732 5093
V 1730 5094 V 1729 5094 V 1727 5095 V 1725 5096 V 1723
5096 V 1721 5097 V 1720 5097 V 1718 5098 V 1716 5098
V 1714 5099 V 1712 5099 V 1711 5100 V 1709 5100 V 1707
5101 V 1705 5101 V 1703 5102 V 1701 5102 V 1700 5103
V 1698 5103 V 1696 5104 V 1694 5104 V 1692 5105 V 1690
5105 V 1689 5105 V 1687 5106 V 1685 5106 V 1683 5107
V 1681 5107 V 1679 5108 V 1678 5108 V 1676 5108 V 1674
5109 V 1672 5109 V 1670 5110 V 1668 5110 V 1666 5110
V 1665 5111 V 1663 5111 V 1661 5112 V 1659 5112 V 1657
5112 V 1655 5113 V 1653 5113 V 1652 5113 V 1650 5114
V 1648 5114 V 1646 5114 V 1644 5115 V 1642 5115 V 1640
5115 V 1638 5116 V 1637 5116 V 1635 5116 V 1633 5117
V 1631 5117 V 1629 5117 V 1627 5117 V 1625 5118 V 1623
5118 V 1621 5118 V 1620 5119 V 1618 5119 V 1616 5119
V 1614 5119 V 1612 5120 V 1610 5120 V 1608 5120 V 1606
5120 V 1604 5121 V 1603 5121 V 1601 5121 V 1599 5121
V 1597 5121 V 1595 5122 V 1593 5122 V 1591 5122 V 1589
5122 V 1587 5122 V 1585 5123 V 1583 5123 V 1582 5123
V 1580 5123 V 1578 5123 V 1576 5123 V 1574 5124 V 1572
5124 V 1570 5124 V 1568 5124 V 1566 5124 V 1564 5124
V 1562 5124 V 1560 5125 V 1559 5125 V 1557 5125 V 1555
5125 V 1553 5125 V 1551 5125 V 1549 5125 V 1547 5125
V 1545 5125 V 1543 5125 V 1541 5125 V 1539 5125 V 1537
5126 V 1536 5126 V 1534 5126 V 1532 5126 V 1530 5126
V 1528 5126 V 1526 5126 V 1524 5126 V 1522 5126 V 1520
5126 V 1518 5126 V 1516 5126 V 1514 5126 V 1512 5126
V 1511 5126 V 1509 5126 V 1507 5126 V 1505 5126 V 1503
5126 V 1501 5126 V 1499 5126 V 1497 5126 V 1495 5126
V 1493 5126 V 1491 5126 V 1489 5125 V 1487 5125 V 1486
5125 V 1484 5125 V 1482 5125 V 1480 5125 V 1478 5125
V 1476 5125 V 1474 5125 V 1472 5125 V 1470 5125 V 1468
5125 V 1466 5124 V 1464 5124 V 1463 5124 V 1461 5124
V 1459 5124 V 1457 5124 V 1455 5124 V 1453 5123 V 1451
5123 V 1449 5123 V 1447 5123 V 1445 5123 V 1443 5123
V 1441 5122 V 1440 5122 V 1438 5122 V 1436 5122 V 1434
5122 V 1432 5121 V 1430 5121 V 1428 5121 V 1426 5121
V 1424 5121 V 1422 5120 V 1421 5120 V 1419 5120 V 1417
5120 V 1415 5119 V 1413 5119 V 1411 5119 V 1409 5119
V 1407 5118 V 1405 5118 V 1403 5118 V 1402 5117 V 1400
5117 V 1398 5117 V 1396 5117 V 1394 5116 V 1392 5116
V 1390 5116 V 1388 5115 V 1387 5115 V 1385 5115 V 1383
5114 V 1381 5114 V 1379 5114 V 1377 5113 V 1375 5113
V 1373 5113 V 1372 5112 V 1370 5112 V 1368 5112 V 1366
5111 V 1364 5111 V 1362 5110 V 1360 5110 V 1359 5110
V 1357 5109 V 1355 5109 V 1353 5108 V 1351 5108 V 1349
5108 V 1347 5107 V 1346 5107 V 1344 5106 V 1342 5106
V 1340 5105 V 1338 5105 V 1336 5105 V 1335 5104 V 1333
5104 V 1331 5103 V 1329 5103 V 1327 5102 V 1326 5102
V 1324 5101 V 1322 5101 V 1320 5100 V 1318 5100 V 1316
5099 V 1315 5099 V 1313 5098 V 1311 5098 V 1309 5097
V 1307 5097 V 1306 5096 V 1304 5096 V 1302 5095 V 1300
5094 V 1298 5094 V 1297 5093 V 1295 5093 V 1293 5092
V 1291 5092 V 1290 5091 V 1288 5090 V 1286 5090 V 1284
5089 V 1283 5089 V 1281 5088 V 1279 5088 V 1277 5087
V 1275 5086 V 1274 5086 V 1272 5085 V 1270 5084 V 1268
5084 V 1267 5083 V 1265 5082 V 1263 5082 V 1262 5081
V 1260 5081 V 1258 5080 V 1256 5079 V 1255 5078 V 1253
5078 V 1251 5077 V 1250 5076 V 1248 5076 V 1246 5075
V 1244 5074 V 1243 5074 V 1241 5073 V 1239 5072 V 1238
5071 V 1236 5071 V 1234 5070 V 1233 5069 V 1231 5069
V 1229 5068 V 1228 5067 V 1226 5066 V 1224 5066 V 1223
5065 V 1221 5064 V 1219 5063 V 1218 5062 V 1216 5062
V 1214 5061 V 1213 5060 V 1211 5059 V 1209 5058 V 1208
5058 V 1206 5057 V 1204 5056 V 1203 5055 V 1201 5054
V 1200 5054 V 1198 5053 V 1196 5052 V 1195 5051 V 1193
5050 V 1192 5049 V 1190 5048 V 1188 5048 V 1187 5047
V 2703 4991 a Fv(Z)p Fy(=3Dp)p Fv(Z)2674 4966 y Fd(?)2694
4945 y(_)2297 4927 y Fl(1)p Fw(7!)p Fx(v)2437 4936 y
Fm(2)2057 4966 y Fg(o)p Ff(o)p 2057 4967 617 4 v eop
%%Page: 11 11
11 10 bop 826 251 a Fq(GALOIS)33=
 b(MODULE)f(CONSTR)n(UCTION)h(AND)f
(CLASSIFICA)-6 b(TION)292 b(11)456 450 y Fz(Let)35 b
Fy(a)684 465 y Fl(2)755 450 y Fu(2)d Fy(F)930 409 y Fw(\002)916
475 y Fl(2)1024 450 y Fz(and)j Fy(b)1257 465 y Fl(1)1329
450 y Fu(2)d Fy(F)1504 409 y Fw(\002)1490 475 y Fl(1)1597
450 y Fz(satisfy)k(\()p Fy(a)1993 465 y Fl(2)2033 450
y Fz(\))2071 465 y Fx(F)2116 474 y Fm(2)2186 450 y Fz(=3D)31
b(\()p Fy(a)p Fz(\))2420 465 y Fx(G)2475 476 y Fs(F)2514
491 y Fm(2)2592 450 y Fz(and)k(\()p Fy(b)2863 465 y Fl(1)2903
450 y Fz(\))2941 465 y Fx(F)2986 474 y Fm(1)3056 450
y Fz(=3D)c(\()p Fy(b)p Fz(\))3280 465 y Fx(G)3335 476 y
Fs(F)3374 491 y Fm(1)3417 450 y Fz(.)456 585 y(Then)f(set)g
Fy(K)939 600 y Fl(1)1006 585 y Fz(=3D)d Fy(F)1172 600 y
Fl(1)1212 585 y Fz(\()1277 545 y Fs(p)1254 507 y Fu(p)p
1337 507 81 4 v 78 x Fy(b)1378 600 y Fl(1)1418 585 y
Fz(\))i(and)g Fy(K)1754 600 y Fl(2)1821 585 y Fz(=3D)f
Fy(F)1988 600 y Fl(2)2027 585 y Fz(\()2093 560 y Fs(p)2069
521 y Fu(p)p 2153 521 91 4 v 2153 585 a Fy(a)2204 600
y Fl(2)2243 585 y Fz(\);)j(these)f(are)f Fv(Z)p Fy(=3Dp)p
Fv(Z)p Fz(-extensions)456 701 y(of)e Fy(F)625 716 y Fl(1)692
701 y Fz(and)h Fy(F)940 716 y Fl(2)979 701 y Fz(,)h(resp)s(ectiv)m(ely)
-8 b(.)45 b(W)-8 b(e)28 b(ma)m(y)h(then)f(iden)m(tify)h(the)f
(left-hand)g Fv(Z)p Fy(=3Dp)p Fv(Z)g Fz(in)456 817=
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(with)h(Gal)o(\()p Fy(K)1480 832 y Fl(1)1520 817 y Fy(=3DF)1632
832 y Fl(1)1671 817 y Fz(\))g(so)f(that)h Fy(\013)2117
832 y Fl(1)2183 817 y Fz(is)g(the)g(surjection)g(of)f(Galois)g(the-)456
934 y(ory)-8 b(.)41 b(Similarly)-8 b(,)29 b(the)e(righ)m(t-hand)f
Fv(Z)p Fy(=3Dp)p Fv(Z)h=
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(Gal\()p Fy(K)3216 949 y Fl(2)3255 934 y Fy(=3DF)3367 949
y Fl(2)3406 934 y Fz(\))456 1050 y(with)d Fy(\013)731
1065 y Fl(2)794 1050 y Fz(the)g(surjection)h(of)e(Galois)g(theory)-8
b(.)41 b(Finally)-8 b(,)26 b(the)e(topmost)g(surjections)456
1166 y Fy(\016)499 1181 y Fl(1)571 1166 y Fz(and)32 b
Fy(\016)803 1181 y Fl(2)876 1166 y Fz(are)g(canonical.)555
1397 y(By)50 b(Lemma)g(7,)j(the)c(surjections)i Fy(\016)1949
1412 y Fl(1)1988 1397 y Fy(\013)2050 1412 y Fl(1)2139
1397 y Fz(and)e Fy(\016)2388 1412 y Fl(2)2428 1397 y
Fy(\013)2490 1412 y Fl(2)2578 1397 y Fz(corresp)s(ond)h(to)f(\014elds)
456 1518 y Fy(F)14 b Fz(\()598 1468 y Fs(p)575 1433 y
Fu(p)p 658 1433 42 4 v 85 x Fy(b)p Fz(\))35 b(and)g Fy(F)14
b Fz(\()1106 1484 y Fs(p)1083 1446 y Fu(p)p 1166 1446
52 4 v 72 x Fy(a)p Fz(\),)35 b(resp)s(ectiv)m(ely)-8
b(.)54 b(No)m(w)35 b(b)s(ecause)i Fy(G)2551 1533 y Fx(F)2641
1518 y Fz(=3D)31 b Fy(G)2825 1533 y Fx(F)2870 1542 y Fm(1)2932
1518 y Fy(?)24 b(G)3082 1533 y Fx(F)3127 1542 y Fm(2)3165
1518 y Fz(,)36 b(there)456 1634 y(exists)30 b(a)e(homomorphism)i
Fy(\014)j Fz(:)28 b Fy(G)1703 1649 y Fx(F)1789 1634 y
Fu(!)g Fy(H)1998 1651 y Fx(p)2034 1632 y Fm(3)2072 1634
y Fz(,)h(and)g(b)s(ecause)h Fy(v)2718 1649 y Fl(1)2786
1634 y Fz(and)f Fy(v)3019 1649 y Fl(2)3087 1634 y Fz(generate)456
1750 y Fy(H)537 1767 y Fx(p)573 1748 y Fm(3)611 1750
y Fz(,)j Fy(\014)38 b Fz(is)c(a)e(surjection.)555 1981
y(Hence)i Fy(H)926 1998 y Fx(p)962 1979 y Fm(3)1033 1981
y Fz(is)f(a)g(Galois)f(group)h(o)m(v)m(er)g Fy(F)47 b
Fz(corresp)s(onding)33 b(to)g(a)f(normal)h(sub-)456 2098
y(group)45 b Fy(H)53 b Fz(of)46 b Fy(G)1081 2113 y Fx(F)1139
2098 y Fz(.)83 b(Consider)47=
 b(the)f(smallest)i(normal)d(subgroup)i
Fy(H)3100 2113 y Fl(1)3185 2098 y Fz(of)e Fy(G)3386 2113
y Fx(F)456 2214 y Fz(con)m(taining)35 b Fy(H)41 b Fz(and)35
b Fy(G)1323 2229 y Fx(F)1368 2238 y Fm(2)1406 2214 y
Fz(.)49 b(Then)35 b(b)m(y)g(the)g(diagram,)g Fy(G)2531
2229 y Fx(F)2589 2214 y Fy(=3DH)2719 2229 y Fl(1)2793 2214
y Fz(is)g(the)f(left-hand)456 2340 y Fv(Z)p Fy(=3Dp)p Fv(Z)p
Fz(,)26 b(whic)m(h)g(corresp)s(onds)g(to)e Fy(F)14 b
Fz(\()1790 2291 y Fs(p)1767 2255 y Fu(p)p 1850 2255 42
4 v 85 x Fy(b)p Fz(\),)26 b(and)f Fy(H)2245 2355 y Fl(1)2284
2340 y Fy(=3DH)31 b Fz(is)25 b Fu(h)p Fy(v)2621 2355 y
Fl(2)2661 2340 y Fy(;)17 b(w)s Fu(i)p Fz(.)39 b(No)m(w)25
b(consider)456 2456 y(the)34 b(smallest)i(normal)e(subgroup)h
Fy(H)1840 2471 y Fl(2)1914 2456 y Fz(of)f Fy(G)2104 2471
y Fx(F)2196 2456 y Fz(con)m(taining)h Fy(H)42 b Fz(and)34
b Fy(G)3063 2471 y Fx(F)3108 2480 y Fm(1)3147 2456 y
Fz(.)48 b(Then)456 2572 y(b)m(y)34 b(the)f(diagram,)g
Fy(G)1244 2587 y Fx(F)1303 2572 y Fy(=3DH)1433 2587 y Fl(2)1505
2572 y Fz(is)g(the)h(righ)m(t-hand)f Fv(Z)p Fy(=3Dp)p=
 Fv(Z)p
Fz(,)h(whic)m(h)g(corresp)s(onds)h(to)456 2689 y Fy(F)14
b Fz(\()598 2655 y Fs(p)575 2617 y Fu(p)p 658 2617 52
4 v 72 x Fy(a)p Fz(\),)32 b(and)h Fy(H)1077 2704 y Fl(2)1116
2689 y Fy(=3DH)40 b Fz(is)33 b Fu(h)p Fy(v)1470 2704 y
Fl(1)1510 2689 y Fy(;)17 b(w)s Fu(i)p Fz(.)555 2920 y(Hence)34
b(\()p Fy(a)p Fz(\))22 b Fu([)h Fz(\()p Fy(b)p Fz(\))28
b(=3D)f(0.)1960 b Fi(\003)555 3296 y Fz(Finally)-8=
 b(,)36
b(w)m(e)h(close)f(with)f(with)h(a)f(companion)h(to)f(Lemma)g(7.)51
b(In)36 b(Lemma)g(9)456 3413 y(b)s(elo)m(w,)c=
 Fy(\031)k
Fz(denotes)c(the)g(canonical)g(homomorphism)h(of)e Fy(G)g
Fz(to)h Fy(G)2896 3428 y Fl(1)2966 3413 y Fz(whic)m(h)h(is)f(an)456
3529 y(iden)m(tit)m(y)i(on)e Fy(G)1028 3544 y Fl(1)1068
3529 y Fz(,)g(and)h(is)g(trivial)g(on)g Fy(G)1918 3544
y Fl(2)1957 3529 y Fz(.)456 3748 y FF(Lemma)48 b(9.)e
Fp(L)-5 b(et)43 b Fy(G)f Fz(=3D)g Fy(G)1485 3763 y Fl(1)1552
3748 y Fy(?)28 b(G)1706 3763 y Fl(2)1788 3748 y Fp(b)-5
b(e)42 b(a)g(fr)-5 b(e)g(e)42 b(pr)-5 b(o)g(duct)43 b(of)f
Fy(G)2752 3763 y Fl(1)2834 3748 y Fp(and)g Fy(G)3108
3763 y Fl(2)3190 3748 y Fp(in)g(the)456 3864 y(c)-5=
 b(ate)g(gory)36
b(of)h(pr)-5 b(o-)p Fy(p)p Fp(-gr)g(oups.)50 b(Supp)-5
b(ose)36 b(that)h Fy(A)2241 3836 y Fu(\030)2242 3868
y Fz(=3D)2350 3864 y Fv(Z)p Fy(=3Dp)p Fv(Z)g=
 Fp(is)g(a)f(factor)h(gr)-5
b(oup)37 b(of)456 3980 y Fy(G)533 3995 y Fl(1)602 3980
y Fp(such)29 b(that)i(the)f(surje)-5 b(ction)29 b Fy(G)1691
3995 y Fl(1)1758 3980 y Fu(!)e Fv(Z)p Fy(=3Dp)p Fv(Z)k
Fp(do)-5 b(es)29 b(not)h(factor)g(thr)-5 b(ough)30 b
Fv(Z)p Fy(=3Dp)3309 3944 y Fl(2)3348 3980 y Fv(Z)p Fp(.)456
4096 y(Then)k(the)h(fol)5 b(lowing)33 b(c)-5 b(ommutative)34
b(diagr)-5 b(am)34 b(c)-5 b(annot)35 b(o)-5 b(c)g(cur:)1725
4364 y Fy(G)1957 4301 y Fx(\031)2097 4330 y Fg(/)p Ff(/)29
b Fg(/)p Ff(/)p 1831 4331 296 4 v 1763 4570 a Fg(\017)p
Ff(\017)1763 4599 y Fg(\017)p Ff(\017)p 1761 4599 4 207
v 2155 4357 a Fy(G)2232 4372 y Fl(1)2214 4588 y Fg(\017)p
Ff(\017)2214 4617 y Fg(\017)p Ff(\017)p 2212 4617 4 218
v 1628 4707 a Fv(Z)p Fy(=3Dp)1792 4671 y Fl(2)1832 4707
y Fv(Z)2105 4680 y Fg(/)p Ff(/)h Fg(/)p Ff(/)p 1927 4682
208 4 v 2164 4715 a Fy(A:)456 5099 y Fp(Pr)-5 b(o)g(of.)41
b Fz(Supp)s(ose)33=
 b(that)f(con)m(trary)h(to)f(our)g(statemen)m(t,)i
(suc)m(h)g(a)e(diagram)g(as)h(the)456 5216 y(ab)s(o)m(v)m(e)h(exists.)
50 b(Then)35=
 b(b)m(y)g(passing)g(to)e(quotien)m(ts)j(b)m(y)f(comm)m
(utator)f(subgroups)p eop
%%Page: 12 12
12 11 bop 456 255 a Fq(12)759 b(J)1340 236 y(\023)1330
255 y(AN)26 b(MIN)1638 236 y(\023)1628 255 y(A)1695 236
y(\024)1686 255 y(C)f(AND)f(JOHN)h(SW)-9 b(ALLO)n(W)456
450 y Fz(w)m(e)33 b(obtain)1673 626 y Fy(G)1750 590 y
Fx(ab)1926 556 y(\031)1969 533 y Fs(ab)2081 585 y Fg(/)p
Ff(/)c Fg(/)p Ff(/)p 1850 587 260 4 v 1673 781 a Fx(\013)1747
838 y Fg(\017)p Ff(\017)1747 867 y Fg(\017)p Ff(\017)p
1745 867 4 212 v 2139 614 a Fy(G)2216 578 y Fx(ab)2216
639 y Fl(1)2243 775 y Fx(\015)2214 856 y Fg(\017)p Ff(\017)2214
885 y Fg(\017)p Ff(\017)p 2212 885 4 218 v 1612 975 a
Fv(Z)p Fy(=3Dp)1776 939 y Fl(2)1816 975 y Fv(Z)1959 906
y Fx(\014)2105 948 y Fg(/)p Ff(/)h Fg(/)p Ff(/)p 1911
950 224 4 v 2164 982 a Fy(A:)456 1151 y Fz(But)g Fy(G)724
1115 y Fx(ab)824 1123 y Fu(\030)825 1155 y Fz(=3D)929 1151
y Fy(G)1006 1115 y Fx(ab)1006 1175 y Fl(1)1096 1151 y
Fu(\002)18 b Fy(G)1268 1115 y Fx(ab)1268 1175 y Fl(2)1370
1151 y Fz(and)31 b(the)g(canonical)g(surjection)h(on)m(to)e
Fy(G)2890 1115 y Fx(ab)2890 1175 y Fl(1)2993 1151 y=
 Fz(is)h(giv)m(en)g
(b)m(y)456 1267 y(the)38 b(pro)5 b(jection)39 b(map.)61
b(Let)38 b Fy(\016)k=
 Fz(b)s(e)d(a)e(splitting)i(map)g(of)e(the)i(pro)5
b(jection)39 b(map.)456 1383 y(Then)33 b Fy(\015)g Fz(=3D)28
b Fy(\014)6 b(\013)q(\016)t Fz(,)31=
 b(con)m(tradicting)j(the)f(h)m(yp)s
(othesis.)1005 b Fi(\003)1305 1699 y Fz(3.)49 b=
 Fj(Pr)n(oof)38
b(of)g(the)g(Theorem)555 1961 y Fz(First)46 b(w)m(e)h(de\014ne)f
(\014elds)h Fy(F)1583 1976 y Fl(0)1623 1961 y Fz(,)i
Fy(F)1762 1976 y Fl(1)1801 1961 y Fz(,)g Fy(F)1940 1976
y Fl(2)1980 1961 y Fz(,)g(and)c Fy(F)60 b=
 Fz(using)46
b(our)f(giv)m(en)i(cardinal)456 2077 y(n)m(um)m(b)s(ers)26
b Fy(d)p Fz(,)f Fy(e)p=
 Fz(,)h(and)e(\007,)i(as)e(w)m(ell)i(as)e(the)g
(prime)h(n)m(um)m(b)s(er)h Fy(p)p Fz(.)40 b(Then)25=
 b(w)m(e)h(de\014ne)
f(the)456 2193 y(cyclic)32 b(Galois)f(extension)h Fy(K)r(=3DF)44
b Fz(of)30 b(degree)i Fy(p)f=
 Fz(and)g(c)m(hec)m(k)i(that)d(the)h
(arithmetic)456 2309 y(in)m(v)-5 b(arian)m(ts)33 b(of)f
Fy(K)r(=3DF)46 b Fz(coincide)34 b(with)f Fy(d)p Fz(,)g
Fy(e)p Fz(,)f(and)h(\007.)456 2593 y(3.1.)48 b FF(Constructing)37
b Fy(F)1394 2608 y Fl(0)1434 2593 y FF(,)g Fy(F)1565
2608 y Fl(1)1605 2593 y FF(,)h Fy(F)1737 2608 y Fl(2)1776
2593 y FF(,)g(and)g Fy(F)14 b FF(.)555 2796 y=
 Fz(If)39
b(\007)f(=3D)h(1)f(then)i(let)f Fy(F)1414 2811 y Fl(0)1492
2796 y Fz(=3D)f Fv(C)p Fz(.)63 b(If)38 b(\007)h(=3D)f(0)h(and)g
Fy(p)f Fz(=3D)g(2,)j(let)e Fy(F)2912 2811 y Fl(0)2990 2796
y Fz(=3D)f Fv(R)p Fz(.)63 b(\(See)456 2912=
 y(Prop)s(osition)33
b(1)f(for)h(alternativ)m(e)h(c)m(hoices)h(in)e(these)h(t)m(w)m(o)g
(cases.\))45 b(If)33 b(\007)28 b(=3D)h(0)j(and)456 3029
y Fy(p)27 b(>)h Fz(2)i(then)h(let)g Fy(F)1137 3044 y
Fl(0)p Fx(;p)1262 3029 y=
 Fz(b)s(e)g(the)g(algebraic)g(extension)h(of)e
Fv(Q)h Fz(of)f(Lemma)h(5.)43 b(In)31 b(the)456 3145=
 y(\014rst)d(t)m(w)m
(o)h(cases)g(w)m(e)g(see)g(trivially)g(that)f Fy(\030)1978
3160 y Fx(p)2045 3145 y Fu(2)g Fy(F)2202 3160 y Fl(0)2242
3145 y Fz(,)h(and)f(in)g(the)h(last)f(case)h Fy(\030)3181
3160 y Fx(p)3248 3145 y Fu(2)f Fy(F)3405 3160 y Fl(0)456
3261 y Fz(b)m(y)33 b(Lemma)g(6.)43 b(Observ)m(e)35 b(that)1077
3574 y(dim)1240 3589 y Ft(F)1281 3597 y Fs(p)1337 3574
y Fy(F)1414 3533 y Fw(\002)1400 3599 y Fl(0)1473 3574
y Fy(=3DF)1599 3527 y Fw(\002)p Fx(p)1585 3599 y Fl(0)1721
3574 y Fz(=3D)1824 3340 y Fo(8)1824 3430 y(>)1824 3460
y(<)1824 3639 y(>)1824 3669 y(:)1913 3438 y Fz(0)p Fy(;)49
b Fz(if)32 b(\007)c(=3D)f(1;)1913 3577 y(1)p Fy(;)49 b
Fz(if)32 b(\007)c(=3D)f(0)p Fy(;)49 b(p)28 b Fz(=3D)f(2;)1913
3717 y(2)p Fy(;)49 b Fz(if)32 b(\007)c(=3D)f(0)p Fy(;)49
b(p)28 b(>)f Fz(2)p Fy(:)3320 3574 y Fz(\(2\))555 3975
y(W)-8 b(e)39 b(next)h(construct)g(the)f(\014eld)h Fy(F)1845
3990 y Fl(1)1885 3975 y Fz(.)62 b(Because)40 b(\007)f(=3D)f(0)g(implies)j
(1)d Fu(\024)h Fy(d)p Fz(,)h(for)456 4091=
 y(either)35
b(c)m(hoice)g(of)f(\007)c Fu(2)h(f)p Fz(0)p Fy(;)17 b
Fz(1)p Fu(g)34 b Fz(there)g(exists)j(a)c(w)m(ell-ordered)j(set)f
Fy(I)2970 4106 y Fl(1)3044 4091 y Fz(suc)m(h)h(that)456
4225 y Fu(j)p Fy(I)527 4240 y Fl(1)566 4225 y Fu(j)20
b Fz(+)h(1)28 b(=3D)f Fy(d)21 b Fz(+)g(2\007.)43 b(Let)32
b(\000)1490 4240 y Fl(1)1557 4225 y Fz(=3D)c Fv(Z)1727
4174 y Fl(\()p Fx(I)1785 4183 y Fm(1)1820 4174 y Fl(\))1727
4257 y(\()p Fx(p)p Fl(\))1883 4225 y Fz(b)s(e)33 b(a)e(direct)i(sum)g
(of)f Fu(j)p Fy(I)2760 4240 y Fl(1)2799 4225 y Fu(j)g
Fz(copies)h(of)e Fv(Z)3322 4241 y Fl(\()p Fx(p)p Fl(\))3417
4225 y Fz(.)456 4356 y(Then)c(\000)765 4371 y Fl(1)830
4356 y Fz(is)g(a)f(linearly)h(ordered)g(ab)s(elian)f(group.)41
b(Finally)27 b(set)g Fy(F)2865 4371 y Fl(1)2932 4356
y Fz(:=3D)h Fy(F)3126 4371 y Fl(0)3165 4356 y Fz(\(\(\000)3302
4371 y Fl(1)3341 4356 y Fz(\)\).)456 4472 y(F)-8=
 b(rom)32
b(Lemmas)h(2)g(and)f(3)h(it)f(follo)m(ws)h(that)g Fy(G)2157
4487 y Fx(F)2202 4496 y Fm(1)2273 4472 y Fz(is)g(a)f(pro-)p
Fy(p)p Fz(-group)f(and)1126 4644 y(dim)1289 4659 y Ft(F)1330
4667 y Fs(p)1386 4644 y Fy(F)1463 4603 y Fw(\002)1449
4668 y Fl(1)1522 4644 y Fy(=3DF)1648 4596 y Fw(\002)p Fx(p)1634
4668 y Fl(1)1770 4644 y Fz(=3D)c(dim)2036 4659 y Ft(F)2077
4667 y Fs(p)2133 4644 y Fy(F)2210 4603 y Fw(\002)2196
4668 y Fl(0)2269 4644 y Fy(=3DF)2395 4596 y Fw(\002)p Fx(p)2381
4668 y Fl(0)2511 4644 y Fz(+)22 b Fu(j)p Fy(I)2680 4659
y Fl(1)2719 4644 y Fu(j)p Fy(:)456 4811 y Fz(Hence)1016
5064 y(dim)1179 5079 y Ft(F)1220 5087 y Fs(p)1276 5064
y Fy(F)1353 5023 y Fw(\002)1339 5089 y Fl(1)1412 5064
y Fy(=3DF)1538 5017 y Fw(\002)p Fx(p)1524 5089 y Fl(1)1660
5064 y Fz(=3D)1763 4830 y Fo(8)1763 4920 y(>)1763 4950
y(<)1763 5129 y(>)1763 5159 y(:)1852 4928 y Fy(d)g Fz(+)g(1)p
Fy(;)97 b Fz(if)33 b(\007)27 b(=3D)h(1;)1852 5067 y Fy(d;)266
b Fz(if)33 b(\007)27 b(=3D)h(0)p Fy(;)49 b(p)27 b Fz(=3D)h(2;)1852
5207 y Fy(d)22 b Fz(+)g(1)p Fy(;)97 b Fz(if)33 b(\007)27
b(=3D)h(0)p Fy(;)49 b(p)27 b(>)h Fz(2)p Fy(:)3320 5064
y Fz(\(3\))p eop
%%Page: 13 13
13 12 bop 826 251 a Fq(GALOIS)33=
 b(MODULE)f(CONSTR)n(UCTION)h(AND)f
(CLASSIFICA)-6 b(TION)292 b(13)555 450 y Fz(Similarly)-8
b(,)35 b(w)m(e)g(construct)g Fy(F)1628 465 y Fl(2)1701
450 y Fz(as)f(follo)m(ws.)47 b(Because)36 b Fy(p)29=
 b(>)h
Fz(2)j(and)h(also)g Fy(p)29 b Fz(=3D)h(2)456 566=
 y(and)37
b(\007)e(=3D)g(1)h(implies)j Fy(e)c(>)g=
 Fz(0,)j(there)f(exists)i(a)e(w)m
(ell-ordered)h(set)g Fy(I)2965 581 y Fl(2)3041 566 y
Fz(suc)m(h)h(that)456 683 y(1)23 b(+)h Fu(j)p Fy(I)699
698 y Fl(2)738 683 y Fu(j)31 b Fz(=3D)h Fy(e)j=
 Fz(in)g(either)h(of)e(the)
i(cases)g Fy(p)31 b(>)h Fz(2)j(or)f(\007)e(=3D)f(1,)36
b Fy(p)31 b Fz(=3D)h(2,)j(and)g Fu(j)p Fy(I)3193 698 y
Fl(2)3232 683 y Fu(j)d Fz(=3D)f Fy(e)456 817 y Fz(in)k(the)h(case)h(\007)
32 b(=3D)g(0,)k Fy(p)d Fz(=3D)f(2.)52 b(Then)37 b(again)e(\000)2181
832 y Fl(2)2253 817 y Fz(=3D)d Fv(Z)2427 766 y Fl(\()p
Fx(I)2485 775 y Fm(2)2520 766 y Fl(\))2427 848 y(\()p
Fx(p)p Fl(\))2587 817 y Fz(is)k(a)f(linearly)i(ordered)456
948 y(ab)s(elian)32 b(group.)44 b(W)-8 b(e)33 b(set)h
Fy(F)1493 963 y Fl(2)1560 948 y Fz(:=3D)28 b Fv(C)p Fz(\(\(\000)1900
963 y Fl(2)1939 948 y Fz(\)\).)44 b(Then)33=
 b(from)g([K,)g(pages)g(3)f
(and)h(4])f(it)456 1064 y(follo)m(ws)g(that)g Fy(G)1063
1079 y Fx(F)1108 1088 y Fm(2)1174 1036 y Fu(\030)1175
1068 y Fz(=3D)1279 1064 y Fv(Z)1345 1028 y Fx(I)1376 1037
y Fm(2)1345 1088 y Fx(p)1415 1064 y=
 Fz(,)g(the)g(top)s(ological)f(pro)s
(duct)h(of)g Fu(j)p Fy(I)2683 1079 y Fl(2)2722 1064 y
Fu(j)f Fz(copies)i(of)f Fv(Z)3245 1079 y Fx(p)3285 1064
y Fz(.)43 b(In)456 1180 y(particular)760 1469 y Fy(e)28
b Fz(=3D)936 1298 y Fo(\()1017 1402 y Fz(dim)1179 1417
y Ft(F)1220 1425 y Fs(p)1277 1402 y Fy(F)1354 1361 y
Fw(\002)1340 1426 y Fl(2)1412 1402 y Fy(=3DF)1538 1355
y Fw(\002)p Fx(p)1524 1426 y Fl(2)1655 1402 y Fz(+)22
b(1)p Fy(;)97 b Fz(if)32 b Fy(p)c(>)g Fz(2)k(or)g Fy(p)c
Fz(=3D)f(2)33 b(and)f(\007)c(=3D)f(1;)1017 1542 y(dim)1179
1557 y Ft(F)1220 1565 y Fs(p)1277 1542 y Fy(F)1354 1500
y Fw(\002)1340 1566 y Fl(2)1412 1542 y Fy(=3DF)1538 1494
y Fw(\002)p Fx(p)1524 1566 y Fl(2)1632 1542 y Fy(;)267
b Fz(if)32 b Fy(p)c Fz(=3D)g(2)p Fy(;)49 b Fz(\007)27 b(=3D)h(0)p
Fy(:)3320 1469 y Fz(\(4\))555 1859 y(F)-8 b(rom)35=
 b(Theorem)h(2)f(w)m
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b Fz(of)35 b(c)m(haracteristic)456 1976 y(zero,)30=
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1991 y Fx(F)1739 2000 y Fm(2)1807 1976 y Fz(is)29=
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3144 y Fx(p)3379 3129 y Fz(\),)456 3245=
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4451 y Fy(=3DF)2567 4404 y Fw(\002)p Fx(p)2553 4476 y Fl(2)2661
4451 y Fy(:)555 4718 y Fz(If)52 b(\007)59 b(=3D)h(1)51
b(then)h Fy(H)1373 4682 y Fl(1)1412 4718 y Fz(\()p Fy(G)1527
4733 y Fx(F)1572 4742 y Fm(0)1611 4718 y Fy(;)17 b Fv(F)1716
4733 y Fx(p)1755 4718 y Fz(\))60 b(=3D)f(0,)e(and)51=
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(Lemma)g(4)f(w)m(e)i(ha)m(v)m(e)456 4834 y(dim)618 4849
y Ft(F)659 4857 y Fs(p)716 4834 y Fz(ann)873 4849 y Fx(F)918
4858 y Fm(1)956 4834 y Fz(\()p Fy(a)p Fz(\))1083 4849
y Fx(G)1138 4860 y Fs(F)1177 4875 y Fm(1)1247 4834 y
Fz(=3D)28 b(1.)555 5040 y(If)33 b Fy(p)27 b(>)h=
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b(=3D)h(0)k(then)h(b)m(y)h(Lemma)f(4,)1138 5208 y(dim)1301
5223 y Ft(F)1342 5231 y Fs(p)1398 5208 y Fz(ann)1555
5223 y Fx(F)1600 5232 y Fm(1)1639 5208 y Fz(\()p Fy(a)p
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5208 y Fz(ann)2451 5223 y Fx(F)2496 5232 y Fm(0)2534
5208 y Fz(\()p Fy(b)p Fz(\))2651 5223 y Fx(F)2696 5232
y Fm(0)2735 5208 y Fy(;)p eop
%%Page: 15 15
15 14 bop 826 251 a Fq(GALOIS)33=
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(CLASSIFICA)-6 b(TION)292 b(15)456 460 y Fz(where)24
b Fy(b)g Fz(w)m(as)g(c)m(hosen)h(so)f(that)f Fy(K)1670
475 y Fl(0)1737 460 y Fz(=3D)28 b Fy(F)1904 475 y Fl(0)1943
460 y Fz(\()2009 411 y Fs(p)1985 375 y Fu(p)p 2069 375
42 4 v 2069 460 a Fy(b)p Fz(\))23=
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(of)f(the)h(factor)456 576 y Fv(Z)522 591 y Fx(p)593
576 y Fz(in)32 b Fy(G)783 540 y Fx(ab)783 601 y(F)828
610 y Fm(0)866 576 y Fz(.)43 b(No)m(w)32 b(b)s(ecause)h(\()p
Fy(b)p Fz(\))1634 591 y Fx(F)1679 600 y Fm(0)1738 576
y Fu([)20 b Fz(\()p Fy(b)p Fz(\))1941 591 y Fx(F)1986
600 y Fm(0)2052 576 y Fz(=3D)28=
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Fy(H)3050 540 y Fl(2)3089 576 y Fz(\()p Fy(G)3204 591
y Fx(F)3262 576 y Fy(;)17 b Fv(F)3367 591 y Fx(p)3406
576 y Fz(\))456 692 y(for)32 b Fy(p)27 b(>)h Fz(2,)1404
811 y(1)g Fu(\024)g Fz(dim)1748 826 y Ft(F)1789 834 y
Fs(p)1846 811 y Fz(ann)2003 826 y Fx(F)2048 835 y Fm(0)2086
811 y Fz(\()p Fy(b)p Fz(\))2203 826 y Fx(F)2248 835 y
Fm(0)2315 811 y Fu(\024)g Fz(2)p Fy(:)456 954 y=
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b(the)h(other)f(hand,)k(let)d Fy(c)i Fu(2)h Fy(F)1795
918 y Fw(\002)1884 954 y Fu(n)30 b Fy(F)2041 918 y Fw(\002)p
Fx(p)2180 954 y Fz(b)s(e)44 b(de\014ned)i(so)f(that)f
Fy(F)3090 969 y Fl(0)3129 954 y Fz(\()3195 920 y Fs(p)3171
882 y Fu(p)p 3255 882 V 3255 954 a Fy(c)p Fz(\))g(is)456
1070 y(con)m(tained)d(in)f(the)g(\014xed)h(\014eld)g(of)e(the)h(factor)
g Fv(Z)p Fy(=3Dp)p Fv(Z)g Fz(in)g Fy(G)2711 1034 y Fx(ab)2711
1095 y(F)2756 1104 y Fm(0)2794 1070 y Fz(.)66=
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b(b)m(y)h(\(2\))o(,)456 1192 y Fu(f)p Fz(\()p Fy(b)p
Fz(\))623 1207 y Fx(F)668 1216 y Fm(0)706 1192 y Fy(;)17
b Fz(\()p Fy(c)p Fz(\))868 1207 y Fx(F)913 1216 y Fm(0)951
1192 y Fu(g)31 b Fz(spans)g Fy(H)1385 1155 y Fl(1)1424
1192 y Fz(\()p Fy(G)1539 1207 y Fx(F)1584 1216 y Fm(0)1623
1192 y Fy(;)17 b Fv(F)1728 1207 y Fx(p)1767 1192 y Fz(\).)43
b(F)-8 b(urthermore,)31 b(since)h(b)m(y)g(Lemma)f(5,)g
Fy(H)3370 1208 y Fx(p)3406 1189 y Fm(3)456 1308 y=
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(quotien)m(t)h(of)f Fy(G)1384 1323 y Fx(F)1429 1332 y
Fm(0)p Fs(;p)1518 1308 y Fz(,)g(\()p Fy(b)p Fz(\))1695
1323 y Fx(F)1740 1332 y Fm(0)1802 1308 y Fu([)23 b Fz(\()p
Fy(c)p Fz(\))2009 1323 y Fx(F)2054 1332 y Fm(0)2120 1308
y Fu(6)p Fz(=3D)29 b(0)k(b)m(y)h([M,)f(Corollary)-8 b(,)34
b(page)f(523)456 1425 y(and)f(Theorem)i(3\(A\)].)f(W)-8
b(e)33 b(conclude)h(that)e(dim)2285 1440 y Ft(F)2326
1448 y Fs(p)2382 1425 y Fz(ann)2539 1440 y Fx(F)2584
1449 y Fm(0)2623 1425 y Fz(\()p Fy(b)p Fz(\))2740 1440
y Fx(F)2785 1449 y Fm(0)2851 1425 y Fz(=3D)c(1.)555 1625
y(Finally)-8 b(,)33 b(if)g Fy(p)27 b=
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b(=3D)h(0)k(then)h(again)f(b)m(y)i(Lemma)f(4)1097 1784
y(dim)1260 1799 y Ft(F)1301 1808 y Fm(2)1356 1784 y Fz(ann)1513
1799 y Fx(F)1558 1808 y Fm(1)1596 1784 y Fz(\()p Fy(a)p
Fz(\))1723 1799 y Fx(G)1778 1810 y Fs(F)1817 1825 y Fm(1)1888
1784 y Fz(=3D)27 b(dim)2154 1799 y Ft(F)2195 1808 y Fm(2)2250
1784 y Fz(ann)2407 1799 y Fx(F)2452 1808 y Fm(0)2490
1784 y Fz(\()p Fu(\000)p Fz(1\))2692 1799 y Fx(F)2737
1808 y Fm(0)2776 1784 y Fy(:)456 1960 y Fz(As)49 b Fy(F)679
1975 y Fl(0)773 1960 y Fz(=3D)54 b Fv(R)p Fz(,)f(dim)1217
1975 y Ft(F)1258 1984 y Fm(2)1313 1960 y Fy(F)1390 1918
y Fw(\002)1376 1984 y Fl(0)1449 1960 y Fy(=3DF)1575 1918
y Fw(\002)p Fl(2)1561 1984 y(0)1724 1960 y Fz(=3D)h(1)48
b(and)h(\()p Fu(\000)p Fz(1\))2359 1975 y Ft(R)2444 1960
y Fu([)33 b Fz(\()p Fu(\000)p Fz(1\))2745 1975 y Ft(R)2852
1960 y Fu(6)p Fz(=3D)55 b(0,)d(yielding)456 2076 y(dim)618
2091 y Ft(F)659 2100 y Fm(2)714 2076 y Fz(ann)871 2091
y Fx(F)916 2100 y Fm(1)955 2076 y Fz(\()p Fy(a)p Fz(\))1082
2091 y Fx(G)1137 2102 y Fs(F)1176 2117 y Fm(1)1246 2076
y Fz(=3D)27 b(0)p Fy(:)555 2284 y Fz(Com)m(bining)34=
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(with)f(\(4\))o(,)g(w)m(e)h(ha)m(v)m(e)g(that)e Fy(e)p
Fz(\()p Fy(K)r(=3DF)14 b Fz(\))27 b(=3D)g Fy(e)p Fz(.)555
2484 y(No)m(w)42=
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Fy(d)p Fz(\()p Fy(K)r(=3DF)14 b Fz(\).)67 b(Again)41=
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2600 y(and)32=
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(case)e(\()p Fy(a)p Fz(\))2301 2615 y Fx(G)2356 2626
y Fs(F)2395 2641 y Fm(2)2465 2600 y Fz(=3D)28 b(0,)1017
2776 y Fy(d)p Fz(\()p Fy(K)r(=3DF)14 b Fz(\))27 b(=3D)g(dim)1648
2791 y Ft(F)1689 2799 y Fs(p)1729 2776 y Fz(\()p Fy(H)1856
2735 y Fl(1)1895 2776 y Fz(\()p Fy(G)2010 2791 y Fx(F)2068
2776 y Fy(;)17 b Fv(F)2173 2791 y Fx(p)2212 2776 y Fz(\))p
Fy(=3D)g Fz(ann)2473 2791 y Fx(F)2532 2776 y Fz(\()p Fy(a)p
Fz(\))2659 2791 y Fx(F)2717 2776 y Fz(\))1382 2936 y(=3D)27
b(dim)1648 2951 y Ft(F)1689 2959 y Fs(p)1729 2936 y Fz(\()p
Fy(H)1856 2895 y Fl(1)1895 2936 y Fz(\()p Fy(G)2010 2951
y Fx(F)2055 2960 y Fm(1)2093 2936 y Fy(;)17 b Fv(F)2198
2951 y Fx(p)2237 2936 y Fz(\))p Fy(=3D)g Fz(ann)2498 2951
y Fx(F)2543 2960 y Fm(1)2581 2936 y Fz(\()p Fy(a)p Fz(\))2708
2951 y Fx(G)2763 2962 y Fs(F)2802 2977 y Fm(1)2845 2936
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b(this)i(last)f(equalit)m(y)i(w)m(e)f(use)g(\(3\))f(together)g(with)h
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3380 y Fx(F)1168 3389 y Fm(1)1206 3365 y Fz(\()p Fy(a)p
Fz(\))1333 3380 y Fx(G)1388 3391 y Fs(F)1427 3406 y Fm(1)1512
3365 y Fz(already)h(ac)m(hiev)m(ed.)74 b(Hence)44 b Fy(d)p
Fz(\()p Fy(K)r(=3DF)14 b Fz(\))42 b(=3D)i Fy(d)e Fz(in)g(all)456
3481 y(cases.)456 3743 y(3.4.)48 b FF(Determining)38
b Fz(\007)g FF(via)g(quotien)m(ts)f(of)h Fy(G)2295 3758
y Fx(F)2353 3743 y FF(.)555 3943 y Fz(It)26=
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(that)e(\007\()p Fy(K)r(=3DF)14 b Fz(\))27 b(=3D)g(\007.)41
b(First)26 b(consider)g(the)g(case)g(\007)i(=3D)g(1.)456
4059 y(Since)i Fy(F)770 4074 y Fl(0)837 4059 y Fz(=3D)e
Fv(C)p Fz(,)i Fy(\030)1113 4074 y Fx(p)1181 4059 y Fz(is)g(a)f
Fy(p)p Fz(th)h(p)s(o)m(w)m(er)g(in)g Fy(F)1980 4074 y
Fl(1)2019 4059 y Fz(,)g(and)g Fy(F)2326 4074 y Fl(1)2365
4059 y Fz(\()2431 4026 y Fs(p)2407 3987 y Fu(p)p 2491
3987 52 4 v 2491 4059 a Fy(a)p Fz(\))f(em)m(b)s(eds)i(in)f(a)f
Fv(Z)p Fy(=3Dp)3306 4023 y Fl(2)3345 4059 y Fv(Z)p Fz(-)456
4182 y(extension)50 b Fy(F)966 4197 y Fl(1)1006 4182
y Fz(\()1071 4148 y Fs(p)1103 4127 y Fm(2)1083 4110 y
Fu(p)p 1166 4110 V 72 x Fy(a)p Fz(\))f(of)f Fy(F)1494
4197 y Fl(1)1534 4182 y Fz(.)92 b(Then)50 b(the)f(surjection)h
Fy(G)2649 4197 y Fx(F)2694 4206 y Fm(1)2787 4182 y Fu(!)55
b Fz(Gal\()p Fy(K)3216 4197 y Fl(1)3255 4182 y Fy(=3DF)3367
4197 y Fl(1)3406 4182 y Fz(\))456 4298 y(factors)44 b(through)h
Fv(Z)p Fy(=3Dp)1330 4262 y Fl(2)1370 4298 y Fv(Z)p Fz(.)81
b(F)-8 b(ollo)m(wing)45 b(the)g(surjection)h(with)f(the)h(canonical)456
4414 y(surjection)h Fy(G)994 4429 y Fx(F)1105 4414 y
Fu(!)k Fy(G)1333 4429 y Fx(F)1378 4438 y Fm(1)1416 4414
y Fz(,)g(w)m(e)c(see)h(that)f Fv(Z)p Fy(=3Dp)2213 4378
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Fy(G)3359 4429 y Fx(F)3417 4414 y Fz(.)456 4530=
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b(b)m(y)f(Lemma)h(7,)e(the)h(surjection)h Fy(G)2196 4545
y Fx(F)2282 4530 y Fu(!)28 b Fv(Z)p Fy(=3Dp)p Fv(Z)33=
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Fz(corresp)s(onds)h(to)e Fy(K)7 b Fz(.)456 4647 y(Hence)41
b Fy(K)r(=3DF)52 b Fz(em)m(b)s(eds)42 b(in)e(a)f Fv(Z)p
Fy(=3Dp)1730 4611 y Fl(2)1770 4647 y Fv(Z)p Fz(-extension)i(of)e
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Fy(K)r(=3DF)14 b Fz(\))27 b(=3D)g(1)h(=3D)f(\007.)555 4980
y(No)m(w)43 b(consider)h(the)f(case)h(\007)g(=3D)h(0)d(and)g
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4980 y Fz(=3D)g Fy(F)3200 4995 y Fl(0)3240 4980 y Fz(\()3305
4931 y Fs(p)3282 4895 y Fu(p)p 3365 4895 42 4 v 85 x
Fy(b)p Fz(\))456 5096 y(do)s(es)c(not)h(em)m(b)s(ed)g(in)g(a)f
Fv(Z)p Fy(=3Dp)1556 5060 y Fl(2)1596 5096 y Fv(Z)p=
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Fy(F)2316 5111 y Fl(0)2356 5096 y Fz(,)i Fy(\030)2468
5111 y Fx(p)2560 5096 y Fy(=3D)-61 b Fu(2)42 b Fy(N)2734
5112 y Fx(K)2794 5121 y Fm(0)2828 5112 y Fx(=3DF)2908 5121
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5121 y Fl(0)3134 5096 y Fz(\).)67 b(\(See)456 5216=
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b(for)g(the)h(de\014nition)h(of)e Fy(K)1568 5231 y Fl(0)1608
5216 y Fz(.\))50 b(Hence)36 b(\()p Fy(b)p Fz(\))2132
5231 y Fx(F)2177 5240 y Fm(0)2239 5216 y Fu([)24 b Fz(\()p
Fy(\030)2410 5231 y Fx(p)2449 5216 y Fz(\))2487 5231
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Fy(H)2998 5179 y Fl(1)3037 5216 y Fz(\()p Fy(G)3152 5231
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5231 y Fx(p)3379 5216 y Fz(\),)p eop
%%Page: 16 16
16 15 bop 456 255 a Fq(16)759 b(J)1340 236 y(\023)1330
255 y(AN)26 b(MIN)1638 236 y(\023)1628 255 y(A)1695 236
y(\024)1686 255 y(C)f(AND)f(JOHN)h(SW)-9 b(ALLO)n(W)456
450 y Fz(and,)32 b(b)m(y)g(Lemma)g(4,)g(\(inf)6 b(\()p
Fy(b)p Fz(\)\))21 b Fu([)f Fz(\()p Fy(\030)1759 465 y
Fx(p)1798 450 y Fz(\))1836 465 y Fx(F)1881 474 y Fm(1)1947
450 y Fu(6)p Fz(=3D)28 b(0)j(in)h Fy(H)2333 414 y Fl(1)2372
450 y Fz(\()p Fy(G)2487 465 y Fx(F)2532 474 y Fm(1)2570
450 y Fy(;)17 b Fv(F)2675 465 y Fx(p)2714 450 y Fz(\))32
b(as)f(w)m(ell.)45 b(Cho)s(ose)456 572 y Fy(b)497 587
y Fl(1)574 572 y Fu(2)37 b Fy(F)754 530 y Fw(\002)740
596 y Fl(1)851 572 y Fz(so)h(that)g(\()p Fy(b)1272 587
y Fl(1)1312 572 y Fz(\))1350 587 y Fx(F)1395 596 y Fm(1)1470
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596 y Fm(0)1940 572 y Fz(and)38 b(set)h Fy(K)2376 587
y Fl(1)2452 572 y Fz(=3D)e Fy(F)2628 587 y Fl(1)2667 572
y Fz(\()2733 531 y Fs(p)2709 494 y Fu(p)p 2793 494 81
4 v 2793 572 a Fy(b)2834 587 y Fl(1)2874 572 y Fz(\).)59
b(Then)40 b Fy(\030)3302 587 y Fx(p)3390 572 y Fy(=3D)-61
b Fu(2)456 689 y Fy(N)534 705 y Fx(K)594 714 y Fm(1)628
705 y Fx(=3DF)708 714 y Fm(1)747 689 y Fz(\()p Fy(K)875
648 y Fw(\002)868 714 y Fl(1)934 689 y Fz(\))28=
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g(Theorem)h(3])e(the)h(\014eld)h(extension)g Fy(K)2868
704 y Fl(1)2908 689 y Fy(=3DF)3020 704 y Fl(1)3087 689
y Fz(do)s(es)f(not)456 809 y(em)m(b)s(ed)43 b(in)f(a)g
Fv(Z)p Fy(=3Dp)1152 772 y Fl(2)1192 809 y Fv(Z)p=
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Fy(F)1916 824 y Fl(1)1955 809 y Fz(.)72 b(Therefore)44
b(the)e(surjection)h Fy(G)3217 824 y Fx(F)3262 833 y
Fm(1)3345 809 y Fu(!)456 935 y Fz(Gal)o(\()p Fy(F)709
950 y Fl(1)749 935 y Fz(\()814 885 y Fs(p)791 850 y Fu(p)p
874 850 42 4 v 85 x Fy(b)p Fz(\))p Fy(=3DF)1065 950 y Fl(1)1105
935 y Fz(\))33 b(do)s(es)g(not)g(factor)g(through)g Fv(Z)p
Fy(=3Dp)2383 898 y Fl(2)2423 935 y Fv(Z)p Fz(.)45 b(F)-8
b(rom)33 b(Lemmas)h(7)f(and)456 1051 y(9)46=
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(surjection)h Fy(G)1646 1066 y Fx(F)1756 1051 y Fu(!)k
Fz(Gal)o(\()p Fy(K)r(=3DF)14 b Fz(\))46=
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(through)456 1167 y Fv(Z)p Fy(=3Dp)620 1131 y Fl(2)659
1167 y Fv(Z)p Fz(.)71 b(Again)41=
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(conclude)h(that)f Fy(\030)2796 1182 y Fx(p)2889 1167
y Fy(=3D)-60 b Fu(2)43 b Fy(N)3065 1183 y Fx(K)q(=3DF)3219
1167 y Fz(\()p Fy(K)3347 1131 y Fw(\002)3406 1167 y Fz(\))456
1283 y(and)32 b(therefore)h(\007\()p Fy(K)r(=3DF)14 b=
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b(=3D)h(0)f(=3D)h(\007.)555 1480 y(Finally)35=
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(\007)31 b(=3D)g(0)k(and)g Fy(p)c Fz(=3D)g(2.)50 b(Because)36
b Fu(\000)p Fz(1)43 b Fy(=3D)-61 b Fu(2)32 b Fy(N)3161
1496 y Ft(C)p Fx(=3D)p Ft(R)3297 1480 y Fz(\()p Fv(C)p
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y Fx(K)2217 1626 y Fm(1)2251 1617 y Fx(=3DF)2331 1626 y
Fm(1)2370 1602 y Fz(\()p Fy(K)2498 1561 y Fw(\002)2491
1626 y Fl(1)2557 1602 y Fz(\).)41 b(By)26 b([A,)h(Theorem)h(3])456
1718 y(the)38 b(surjection)h Fy(G)1159 1733 y Fx(F)1204
1742 y Fm(1)1279 1718 y Fu(!)d Fz(Gal)o(\()p Fy(K)1688
1733 y Fl(1)1727 1718 y Fy(=3DF)1839 1733 y Fl(1)1879 1718
y Fz(\))h(do)s(es)i(not)e(factor)h(through)f Fv(Z)p Fy(=3D)p
Fz(4)p Fv(Z)p Fz(.)60 b(As)456 1834 y(in)39=
 b(the)h(previous)g(case,)i
(b)m(y)e(Lemmas)h(7)e(and)g(9)g(w)m(e)h(see)g(that)f
Fy(K)r(=3DF)53 b Fz(do)s(es)39 b(not)456 1950 y(em)m(b)s(ed)30
b(in)e(a)h Fv(Z)p Fy(=3D)p Fz(4)p Fv(Z)p Fz(-extension)h(of)e
Fy(F)14 b Fz(.)42 b(Again)29=
 b(b)m(y)g([A,)h(Theorem)g(3])e(w)m(e)i
(see)g(that)456 2067 y Fu(\000)p Fz(1)39 b Fy(=3D)-60 b
Fu(2)28 b Fy(N)782 2082 y Fx(K)q(=3DF)936 2067 y Fz(\()p
Fy(K)1064 2031 y Fw(\002)1123 2067 y Fz(\))33 b(and)f(therefore)h
(\007\()p Fy(K)r(=3DF)14 b Fz(\))27 b(=3D)h(0)f(=3D)h(\007.)555
2263=
 y(Hence)i(w)m(e)f(ha)m(v)m(e)h(c)m(hec)m(k)m(ed)h(in)e(all)f
(cases)h(that)g(\007\()p Fy(K)r(=3DF)14 b Fz(\))26=
 b(=3D)i(\007,)h(and)f
(our)g(pro)s(of)456 2380 y(of)k(Theorem)i(1)e(is)h(no)m(w)g(complete.)
1601 b Fi(\003)456 2550 y FF(Remark.)45 b=
 Fz(In)c([EH,)h(Lemma)e(1.2)g
(and)g(Prop)s(osition)g(1.3])f(Efrat)h(and)g(Haran)456
2667=
 y(construct)c(some)g(\014elds)h(with)f(prescrib)s(ed)h(absolute)f
(Galois)f(groups)h(together)456 2783=
 y(with)e(some)g(b)s(ounds)g(on)f
(the)h(transcendence)i(degrees)f(of)e(these)h(\014elds.)48
b(These)456 2899 y(b)s(ounds,)33=
 b(together)g(with)g(the)g(replacemen)m
(t)i(of)d Fv(C)g Fz(b)m(y)i(an)e(algebraic)h(closure)h(of)456
3015 y Fv(Q)d Fz(and)h(of)f Fv(R)g=
 Fz(b)m(y)h(a)f(real-closed)i
(algebraic)e(n)m(um)m(b)s(er)i(\014eld)f Fy(R)h=
 Fz(in)e(the)h(case)g
(when)456 3132 y Fy(p)27 b Fz(=3D)h(2)k(and)h(\007)27=
 b(=3D)h(0)k(in)h(our)
f(pro)s(of)g(ab)s(o)m(v)m(e,)i(yield)f(the)g(follo)m(wing)g(prop)s
(osition.)456 3302 y FF(Prop)s(osition)42 b(1.)h Fp(Supp)-5
b(ose)38 b(that)h Fy(p)f Fp(is)g(a)g(prime)g(numb)-5
b(er,)39 b(and)e(let)i Fy(d)p Fp(,)g Fy(e)p Fp(,)g(and)456
3419 y Fz(\007)27 b Fu(2)h(f)p Fz(0)p Fy(;)17 b Fz(1)p
Fu(g)34 b Fp(b)-5 b(e)34 b(c)-5 b(ar)g(dinal)33 b(numb)-5
b(ers)34 b(such)g(that)h(if)f Fz(\007)28 b(=3D)f(0)34 b
Fp(then)h Fz(1)27 b Fu(\024)h Fy(d)p Fp(,)34 b(if)g Fy(p)28
b(>)g Fz(2)456 3535 y Fp(then)38 b Fz(1)e Fu(\024)g Fy(e)p
Fp(,)k(and)e(if)h Fy(p)c Fz(=3D)h(2)i Fp(and)h Fz(\007)c(=3D)g(1)k
Fp(then)g Fz(1)c Fu(\024)h Fy(e)p Fp(.)58 b(Then)38 b(ther)-5
b(e)39 b(exists)g(a)456 3651 y(\014eld)g Fy(F)54 b Fp(c)-5
b(ontaining)39 b Fv(Q)p Fz(\()p Fy(\030)1426 3666 y Fx(p)1465
3651 y Fz(\))h Fp(and)g(a)g(cyclic)f(Galois)h(extension)f
Fy(K)47 b Fp(of)40 b(de)-5 b(gr)g(e)g(e)39 b Fy(p)456
3767 y Fp(over)34 b Fy(F)49 b Fp(such)34 b(that)851 3924
y Fy(e)p Fz(\()p Fy(K)r(=3DF)14 b Fz(\))27 b(=3D)g Fy(e;)117
b(d)p Fz(\()p Fy(K)r(=3DF)14 b Fz(\))26 b(=3D)i Fy(d;)116
b Fp(and)134 b Fz(\007\()p Fy(K)r(=3DF)14 b Fz(\))27 b(=3D)g(\007)p
Fy(:)456 4081 y Fp(Mor)-5 b(e)g(over)1220 4197 y Fz(tr)p
Fy(:)17 b Fz(deg)r(\()p Fy(F)8 b(=3D)p Fv(Q)p Fz(\))28
b Fu(\024)g Fz(1)22 b(+)g(max)p Fu(f)p Fy(e;)17 b(d)22
b Fz(+)g(1)p Fu(g)p Fy(:)456 4334 y Fp(In)41 b(p)-5 b(articular)41
b(if)h Fy(d;)17 b(e)39 b Fu(2)i Fv(N)27 b Fu([)h(f)p
Fz(0)p Fu(g)p Fp(,)42 b(ther)-5 b(e)42=
 b(exists)f(a)h(Galois)f(cyclic)g
(extension)456 4450 y Fy(K)r(=3DF)50 b Fp(of)37 b(de)-5
b(gr)g(e)g(e)36 b Fy(p)h Fp(with)g(pr)-5 b(escrib)g(e)g(d)37
b(invariants)f Fy(d)p Fp(,)h Fy(e)p Fp(,)h(and)f=
 Fz(\007)g
Fp(of)g(\014nite)f(tr)-5 b(an-)456 4566 y(sc)g(endenc)g(e)33
b(de)-5 b(gr)g(e)g(e)34 b(over)h(its)g(prime)f(\014eld)g
Fv(Q)p Fp(.)1393 4845 y Fz(4.)48 b Fj(A)m(ckno)n(wledgements)555
5099 y Fz(W)-8 b(e)46=
 b(w)m(ould)g(lik)m(e)g(to)e(thank)i(the)f
(organizers)h(of)e(the)i(MSRI)f(programs)g(on)456 5216
y(Galois)26=
 b(theory)i(and)f(the)g(MSRI)g(sta\013)g(for)f(giving)i(us)f
(the)g(opp)s(ortunit)m(y)h(to)e(meet)p eop
%%Page: 17 17
17 16 bop 826 251 a Fq(GALOIS)33=
 b(MODULE)f(CONSTR)n(UCTION)h(AND)f
(CLASSIFICA)-6 b(TION)292 b(17)456 450 y Fz(and)28=
 b(to)h(b)s(egin)g
(our)f(collab)s(oration)g(in)h(the)g(F)-8 b(all)29 b(of)f(1999.)41
b(W)-8 b(e)29 b(are)g(also)g(grateful)456 566 y(to)24
b(A.)h(W)-8 b(adsw)m(orth)26=
 b(for)f(stim)m(ulating)h(con)m(v)m
(ersations.)43 b(The)26 b(\014rst)g(author)e(is)i(v)m(ery)456
683 y(appreciativ)m(e)35=
 b(of)f(the)h(kind)g(assistance)h(of)d(Ron)h
(Hemphill,)i(manager)e(of)g(Iv)m(est)456 799=
 y(Prop)s(erties)44
b(Limited)g(\(London,)i(Canada\),)g(for)c(ha)m(ving)i(pro)m(vided)h
(excellen)m(t)456 915 y(w)m(orking)33 b(conditions.)1660
1950 y Fj(References)456 2227 y FE([A])133 b(A.)28=
 b(Alb)r(ert.)g
Fr(On)h(cyclic)i(\014elds)p FE(.)d(T)-7 b(rans.)27=
 b(Amer.)h(Math.)f
(So)r(c.)h Fc(37)f FE(\(1935\),)f(454{462.)456 2326 y([A)-7
b(T])80 b(E.)26 b(Artin)g(and)h(J.)f(T)-7 b(ate.)26=
 b
Fr(Class)j(\014eld)h(the)l(ory)p=
 FE(.)d(Second)f(edition.)g(Redw)n(o)r
(o)r(d)g(Cit)n(y)-7 b(,)27 b(CA:)697 2426 y(Addison-W)-7
b(esley)27 b(Adv)-5 b(anced)28 b(Bo)r(ok)e(Program,)f(1990.)456
2526 y([E1])96 b(I.)22 b(Efrat.)g Fr(Pr)l(o-)p=
 FB(p)p
Fr(-Galois)k(gr)l(oups)f(of)h(algebr)l(aic)h(extensions)d(of)i
Fb(Q)p FE(.)c(J.)g(Num)n(b)r(er)h(Theory)697 2625 y=
 Fc(64)k
FE(\(1997\),)f(84{99.)456 2725 y([E2])p 697 2725 250
4 v 359 w(.)38 b Fr(Finitely)j(gener)l(ate)l(d)g(pr)l(o-)p
FB(p)p Fr(-absolute)f(Galois)i(gr)l(oups)e(over)h(glob)l(al)g(\014elds)
p FE(.)697 2825 y(J.)27 b(Num)n(b)r(er)h(Theory)f=
 Fc(77)h
FE(\(1999\),)e(83{96.)456 2924 y([EH])76 b(I.)38=
 b(Efrat)f(and)h(D.)h
(Haran.)f=
 Fr(On)g(Galois)k(gr)l(oups)d(over)i(pythagor)l(e)l(an)g(and)f
(semi-r)l(e)l(al)697 3024 y(close)l(d)30=
 b(\014elds)p
FE(.)f(Israel)d(J.)h(Math.)h Fc(85)f FE(\(1994\),)g(no.)g(1-3,)g
(57{78.)456 3123 y([JP])95 b(C.)22 b(U.)h(Jensen)g(and)f(A.)h(Prestel.)
f=
 Fr(R)l(e)l(alization)k(of)g(\014nitely)f(gener)l(ate)l(d)h(pr)l
(o\014nite)f(gr)l(oups)697 3223 y(by)34=
 b(maximal)h(ab)l(elian)g
(extensions)e(of)i(\014elds)p FE(.)d(J.)g(reine)g(angew.)f(Math.)h
Fc(447)g FE(\(1994\),)697 3323 y(201{218.)456 3422 y([K])130
b(J.)63 b(Ko)r(enigsmann.)g=
 Fr(Solvable)h(absolute)g(Galois)g(gr)l
(oups)f(ar)l(e)h(metab)l(elian)p FE(.)h(In-)697 3522
y(v)n(en)n(t.)27 b(Math.)h Fc(144)e FE(\(2001\),)h(1{22.)456
3622 y([La])101 b(S.)20 b(Lang.)g=
 Fr(A)n(lgebr)l(a)p
FE(.)h(Revised)f(third)h(edition.)f(Graduate)g(T)-7=
 b(exts)20
b(in)h(Mathematics)f Fc(211)p FE(.)697 3721 y(New)27
b(Y)-7 b(ork:)37 b(Springer-V)-7 b(erlag,)25 b(2002.)456
3821 y([M])119 b(R.)44 b(Massy)-7 b(.)42 b Fr(Construction)j(de)g
FB(p)p Fr(-extensions)f(Galoisiennes)i(d'un)f(c)l(orps)g(de)g(c)l(ar-)
697 3920 y(act)n(\023)-40 b(eristique)30 b(di\013)n(\023)-40
b(er)l(ente)30 b(de)h FB(p)p FE(.)c(J.)g(Algebra)g=
 Fc(109)g
FE(\(1987\),)f(no.)i(2,)f(508{535.)456 4020 y([MS])73
b(J.)36 b(Min\023)-42 b(a)n(\024)j(c)36=
 b(and)h(J.)f(Sw)n(allo)n(w.)g
Fr(Galois)k(mo)l(dule)f(structur)l(e)d(of)k FB(p)p=
 Fr(th-p)l(ower)e
(classes)i(of)697 4120 y(extensions)29 b(of)i(de)l(gr)l(e)l(e)f
FB(p)p FE(.)d(T)-7 b(o)27 b(app)r(ear,)g(Israel)g(J.)g(Math.)456
4219 y([N])133 b(J.)35 b(Neukirc)n(h.)g Fr(F)-6=
 b(r)l(eie)38
b(Pr)l(o)l(dukte)f(pr)l(o-end)t(licher)j(Grupp)l(en)d(und)g(ihr)l(e)h
(Kohomolo)l(gie)p FE(.)697 4319 y(Arc)n(h.)27 b(Math.)h(\(Basel\))f
Fc(22)g FE(\(1971\),)f(337{357.)456 4419 y([Rib])65 b(P)-7
b(.)38 b(Rib)r(en)n(b)r(oim.)h=
 Fr(Some)h(examples)g(of)h(value)l(d)g
(\014elds)p FE(.)e(J.)f(of)g(Algebra)g Fc(173)g FE(\(1995\),)697
4518 y(668{678.)456 4618 y([Ris])78 b(L.)34=
 b(Rib)r(es.)i
Fr(Intr)l(o)l(duction)f(to)i(pr)l(o\014nite)f(gr)l(oups)h(and)g(Galois)
h(c)l(ohomolo)l(gy)p FE(.)f(Queen's)697 4717=
 y(P)n(ap)r(ers)31
b(in)j(Pure)f(and)g(Applied)h(Mathematics)f Fc(24)p=
 FE(.)h(Kingston,)g
(Canada:)47 b(Queen's)697 4817 y(Univ)n(ersit)n(y)-7
b(,)26 b(1970.)456 4917 y([S])149 b(J.-P)-7 b(.)31=
 b(Serre.)h
Fr(L)l(o)l(c)l(al)i(Fields)p FE(.)g(T)-7 b(rans.)32=
 b(Marvin)f(Ja)n(y)g
(Green)n(b)r(erg.)g(Graduate)h(T)-7 b(exts)32 b(in)697
5016 y(Mathematics)27 b Fc(67)p FE(.)g(New)h(Y)-7=
 b(ork-Berlin:)35
b(Springer-V)-7 b(erlag,)25 b(1979.)456 5116 y([W])110
b(A.)24 b(W)-7 b(adsw)n(orth.)23 b FB(p)p Fr(-Henselian)j(\014elds:)37
b FB(K)6 b Fr(-the)l(ory,)28=
 b(Galois)g(c)l(ohomolo)l(gy,)h(and)e(gr)l
(ade)l(d)697 5216 y(Witt)i(rings.)f FE(P)n(aci\014c)f(J.)g(Math.)h
Fc(105)f FE(\(1983\),)f(473{496.)p eop
%%Page: 18 18
18 17 bop 456 255 a Fq(18)759 b(J)1340 236 y(\023)1330
255 y(AN)26 b(MIN)1638 236 y(\023)1628 255 y(A)1695 236
y(\024)1686 255 y(C)f(AND)f(JOHN)h(SW)-9 b(ALLO)n(W)555
450 y FC(Dep)j(ar)g(tment)51 b(of)e(Ma)-6 b(thema)g(tics,)55
b(Middlesex)49 b(College,)102 b(University)49 b(of)456
550 y(Western)31 b(Ont)-6 b(ario,)32 b(London,)d(Ont)-6
b(ario)63 b(N6A)31 b(5B7)62 b(CANAD)n(A)555 732 y=
 Fr(E-mail)31
b(addr)l(ess)7 b FE(:)38 b Fa(minac@uwo.ca)555 985 y
FC(Dep)-6 b(ar)g(tment)27 b(of)e(Ma)-6 b(thema)g(tics,)27
b(D)n(a)-7 b(vidson)23 b(College,)j(Bo)n(x)e(7046,)h(D)n(a)-7
b(vidson,)456 1084 y(Nor)h(th)31 b(Car)n(olina)62 b(28035-7046)g(USA)
555 1267 y Fr(E-mail)31 b(addr)l(ess)7 b FE(:)38 b=
 Fa(joswallow@davids)
o(on)o(.ed)o(u)p eop
%%Trailer
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userdict /end-hook known{end-hook}if
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