CONSTRUCTION AND CLASSIFICATION OF SOME GALOIS MODULES

JÁN MINÁČ^{∗†} AND JOHN SWALLOW^{‡}

Abstract. In our previous paper we describe the Galois module structures of pth-power class groups K^{×}/K^{×p}, where K/F is a cyclic extension of degree p over a field F containing a primitive pth 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^{×}/K^{×p}.

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

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

• d = d(K/F) := dim_{𝔽_{p}} F^{×}/N(K^{×}),

• e = e(K/F) := dim_{𝔽_{p}} N(K^{×})/F^{×p}, and

• Υ(K/F) := 1 or 0 according to whether ξ_{p} ∈ N(K^{×}) or not.

Now if G = ℤ/pℤ, then J may be considered an 𝔽_{p}[G]-module via any isomorphism G ≅ 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 1 below. Corollaries 1 and 2 of Theorem 1 describe all modules J in an explicit way.

Date: April 14, 2003.

∗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.

†Supported by the Mathematical Sciences Research Institute, Berkeley.

‡Research supported in part by National Security Agency grant MDA904-02-1-0061.

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

Theorem 1. Let p be a prime number. For arbitrary cardinal numbers d, e, and for Υ ∈ {0, 1}, there exists a cyclic field extension K/F of degree p containing a primitive pth root of unity with invariants (d, e, Υ) if and only if

• if Υ = 0, then 1 ≤ d,

• if p > 2 then 1 ≤ e, and

• if p = 2 and Υ = 1 then 1 ≤ e.

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

Corollary 1. Let p > 2 be a prime number, and let G be a cyclic group of order p. Then an 𝔽_{p}[G]-module J is realizable as an 𝔽_{p}[Gal(K/F)]- module K^{×}/K^{×p} for some cyclic G-extension K/F such that F contains a primitive pth root of unity if and only if there exist cardinal numbers d, e, and Υ ∈ {0, 1} such that

(i) If Υ = 0, then 1 ≤ d; (ii) 1 ≤ e; and (iii)

J = (⨁_{j∈𝔎_{1}} M_{1,j}) ⨁ (⨁_{j∈𝔎_{2}} M_{2,j}) ⨁ (⨁_{j∈𝔎_{p}} M_{p,j}) ,

where

(1) |𝔎_{1}| + 1 = 2Υ + d,

(2) |𝔎_{2}| = 1 − Υ, and

(p) |𝔎_{p}| + 1 = e.

The invariants d, e, and Υ determine the module J uniquely.

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

Corollary 2. 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^{×}/K^{×2} for some quadratic extension K/F with its arithmetic invariants d(K/F), e(K/F), and Υ(K/F) coinciding with d, e, and Υ ∈ {0, 1}, respectively, if and only if d, e, and Υ satisfy the conditions below.

• if Υ = 0, then 1 ≤ d,

• if Υ = 1, then 1 ≤ e.

In this case

J = (⨁_{j∈𝔎_{1}} M_{1,j}) ⨁ (⨁_{j∈𝔎_{2}} M_{2,j})

where

(1) |𝔎_{1}| + 1 = 2Υ + d and

(2) |𝔎_{2}| + Υ = e.

Moreover, such a module J is determined uniquely by the invariants 2Υ + d and by e − Υ if e is finite and by e alone if e is infinite.

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

In the case p = 2, we see from [MS, Corollary 2] again that the arithmetic invariants d, e, and Υ determine our module 𝔽_{2}[G], but two isomorphic 𝔽_{2}[G]-modules may have different arithmetic invariants; see [MS, Corollary 3]. 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}^{×}/K_{1}^{×2} and K_{2}^{×}/K_{2}^{×2} are isomorphic to a trivial 𝔽_{2}[G]-module 𝔽_{2}, but their arithmetic invariants (d_{i}, e_{i}, Υ_{i}) are (0, 1, 1) for i = 1 and (2, 0, 0) for i = 2.

1. Notation and Strategy

In all that follows F denotes a field, F^{×} = F \ {0} the multiplicative group of F, p a prime number, and F^{×}/F^{×p} the group of pth-power classes of F. For each f ∈ F^{×} we denote by [f] the class of f in F^{×}/F^{×p}. For each subset A of F^{×} we denote by [A] the set of classes {[a] | a ∈ A} and by ⟨[A]⟩ the subgroup of F^{×}/F^{×p} generated by [A].

We denote by ξ_{p} a primitive pth root of unity in F. (Some fields will be assumed to contain such a primitive pth root; for the other fields in this paper, we will prove that a primitive pth root is contained in the field.) Observe that our assumption that there exists a primitive pth root of unity implies that char(F) ≠ 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} , 𝔽_{p}) are Galois cohomology groups of F with coefficients in 𝔽_{p}. Since all absolute Galois groups will be pro-p-groups, all considered 𝔽_{p} modules are trivial. Finally, let |B| be the cardinal number of a set B.

First observe that the conditions on d, e, and Υ listed in our theorem above are necessary:

(1) If Υ = 0, then ξ_{p} ∉ N_{K/F} (K^{×}) and hence

d = dim_{𝔽_{p}} F^{×}/N_{K/F} (K^{×}) ≥ 1;

(2) If p > 2 and K = F (\sqrt[p]{a}) for a suitable a ∈ F^{×} \ F^{×p}, then a = N_{K/F} (\sqrt[p]{a}) and hence

e = dim_{𝔽_{p}} N_{K/F} (K^{×}) > 0.

(3) If p = 2, Υ = 1, and K = F (\sqrt{a}) for a suitable a ∈ F^{×} \ F^{×2}, then −1 ∈ N_{K/F} (K^{×}) since Υ = 1. Consequently a ∈ N_{K/F} (K^{×}), and thus 1 ≤ e.

Therefore in order to prove Theorem 1 when p > 2, it is sufficient to show, for each cardinal numbers d, e as above, for each Υ ∈ {0, 1}, and for each prime number p > 2, the existence of a field F such that:

• F contains a primitive pth root ξ_{p};

• F^{×}/F^{×p} decomposes intro a direct sum of subgroups

F^{×}/F^{×p} = D ⊕ ⟨[a]⟩ ⊕ E,

where dim_{𝔽_{p}}(⟨[a]⟩ ⊕ E) = e and, setting K = F (\sqrt[p]{a}),

(1) [N_{K/F} (K^{×})] = ⟨[a]⟩ ⊕ E;

(2) dim_{𝔽_{p}}(F^{×}/N_{K/F} (K^{×})) = dim_{𝔽_{p}} D = d; and

(3) Υ = 0 if and only if ξ_{p} ∉ N_{K/F} (K^{×}).

In the case p = 2 and Υ = 1 we use the same conditions as above, and if p = 2 and Υ = 0 we require instead that e = dim_{𝔽_{2}} E and d = dim_{𝔽_{2}}(D ⊕ ⟨[a]⟩). The latter condition is imposed because −1 ∉ N_{F (\sqrt{a})/F} (F (\sqrt{a})^{×}) if and only if a ∉ N_{F (\sqrt{a})/F} (F (\sqrt{a})^{×}).

Our strategy is to interpret the required conditions on F^{×}/F^{×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.

Theorem 2. (Efrat-Haran; see [EH, Proposition 1.3]) Let F_{1}, …, F_{n} be fields of equal characteristic such that G_{F_{1}}, …, G_{F_{n}} are pro-p-groups. Then there exists a field F of the same characteristic such that

G_{F} ≅ G_{F_{1}} ⋆ ⋯ ⋆ G_{F_{n}},

where the product is free in the category of pro-p-groups.

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}.

2. Lemmas

2.1. Valued fields F_{1} with prescribed residue field F_{0} and valuation group Γ.

Let v be a valuation on a field F_{1}, written additively. Then we denote by A_{v} the valuation ring {f ∈ F_{1} | v(f) ≥ 0}; by M_{v} the unique maximal ideal {f ∈ A_{v} | v(f) > 0} of A_{v}; by F_{v} the residue field A_{v}/M_{v} of v; by Γ the valuation group v(F_{1}^{×}) of v; and by U the group A_{v} \ M_{v} of units of v.

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

Lemma 1. Let F_{1} be a valued field with valuation v, valuation group Γ = v(F_{1}^{×}), and group of units U. For each prime p ≠ char(F_{1}) there exists an isomorphism

φ : F_{1}^{×}/F_{1}^{×p} ⟶ U/U^{p} ⊕ Γ/pΓ.

In particular

dim_{𝔽_{p}} F_{1}^{×}/F_{1}^{×p} = dim_{𝔽_{p}} U/U^{p} + dim_{𝔽_{p}} Γ/pΓ.

It is well-known that for each field F_{0} and for each totally ordered abelian group Γ, there exists a field F_{1} with a valuation v : F_{1} → Γ ∪ {∞} such that the residue field F_{v} is isomorphic to F_{0} and the valuation group is Γ.

In order to construct such a field, set

F_{1} = F_{0}((Γ)) := {f : Γ → F_{0} | supp(f) is well-ordered}.

Thus a typical element f ∈ F_{1} can be written as a formal sum f = ∑_{g∈Γ} a_{g}t^{g} such that the set supp(f) := {g ∈ G | a_{g} ≠ 0} is a wellordered subset of G. The valuation v on f is defined as: v(0) = ∞ and v(f) = min supp(f) for f ≠ 0. An important property of the valued field F_{1} as above is the fact that it is henselian. (See for example [Rib, (1.3)].) In what follows we will identify F_{v} with F_{0}. We will also assume that char F_{0} ≠ p.

We will be particularly interested in controlling the pth-power classes of such a field. To do so, we choose particular groups Γ for our valuation groups. These groups will be direct sums of

ℤ_{(p)} := \{\frac{a}{b} ∈ ℚ | a, b ∈ ℤ, b ≠ 0; if a ≠ 0 then (a, b) = 1, p ∤ b\} .

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

Γ = ℤ_{(p)}^{(I)} := {γ : I → ℤ_{(p)} | | supp(γ)| < ∞} .

Thus Γ is a direct sum of |I| copies of ℤ_{(p)}. Observe that ℤ_{(p)} carries a natural ordering induced from ℚ, and then we may order Γ lexicographically, as follows. Let γ_{1} ≠ γ_{2} ∈ Γ. Then γ_{1} < γ_{2} if and only if γ_{1}(i) < γ_{2}(i) for the least element i ∈ I such that γ_{1}(i) ≠ γ_{2}(i). Then Γ is a linearly ordered abelian group. Recall that each non-empty set can be well-ordered. (See [La, Appendix 2, Theorem 4.1].)

We choose Γ as above because G_{F_{0}((Γ))} will be pro-p (see Lemma 3 below) and because we may control the pth-power classes with the following lemma. This well-known lemma follows from Lemma 1 and the fact that the valued field F_{1} is henselian. It is also an immediate consequence of [W, Lemma 1.4]. Therefore we shall omit its proof.

Lemma 2. Let F_{1} = F_{0}((Γ)) as above. Then

dim_{𝔽_{p}} F_{1}^{×}/F_{1}^{×p} = dim_{𝔽_{p}} F_{0}^{×}/F_{0}^{×p} + dim_{𝔽_{p}} Γ/pΓ
= dim_{𝔽_{p}} F_{0}^{×}/F_{0}^{×p} + |I|.

Finally, we record a criterion for G_{F_{1}} being pro-p:

Lemma 3. Let F_{1} = F_{0}((Γ)) as above with char(F_{0}) = 0 and G_{F_{0}} pro-p. Then G_{F_{1}} is pro-p as well.

Proof. From basic valuation theory, nicely summarized in [K, pages 3 and 4], and the fact that F_{1} above is henselian,

G_{F_{1}} ≅ T ⋊ G_{F_{0}},

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 ≅ ℤ_{p}^{I}, the topological product of |I| copies of ℤ_{p}. In particular, if G_{F_{0}} is a pro-p-group, so is G_{F_{1}}. □

2.2. H^{2}(G_{F_{1}}, 𝔽_{p}) for henselian valued fields F_{1}.

Now we study H^{2}(G_{F_{1}}, 𝔽_{p}) for our henselian valued fields F_{1}. The next lemma, taken from [W], will be used in the proof of Theorem 1 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 Γ = v(F_{1}^{×}). Let F_{1}^{nr} denote the maximal unramified extension of F_{1} in its separable closure F_{1}^{s}. Then G_{F_{0}} ≅ G_{F_{1}}/ Gal(F_{1}^{s}/F_{1}^{nr}). Therefore, after identifying these groups, we have the inflation map

inf = inf _{F_{0}}^{F_{1}} : H^{∗}(G_{F_{0}}, 𝔽_{p}) ⟶ H^{∗}(G_{F_{1}}, 𝔽_{p}).

(See [W, page 483].)

Moreover, from basic Kummer theory we have the canonical isomorphism

φ_{F} : F^{×}/F^{×p} ⟶ H^{1}(G_{F} , 𝔽_{p}),

as well as the corresponding canonical isomorphisms φ_{F_{i}}, i = 1, 2. We will denote by (f)_{F} or (f_{i})_{F_{i}} the images φ_{F} ([f]) or φ_{F_{i}}([f_{i}]). If the context is clear we will omit the subscript.

Assume next that {π_{j}, j ∈ J } is a set of elements of F_{1}^{×} such that their images in Γ/pΓ form a basis of Γ/pΓ over 𝔽_{p}. Then we have the following lemma, obtained as a special case of a theorem of Wadsworth.

Lemma 4. [W, Theorem 3.6, page 483]. Let F_{1} and F_{0} be as above. Then

H^{2}(G_{F_{1}}, 𝔽_{p}) = inf(H^{2}(G_{F_{0}}, 𝔽_{p})) ⊕_{j∈J} (inf(H^{1}(G_{F_{0}}, 𝔽_{p})) ∪ (π_{j}))
⊕_{{j_{1},j_{2}}⊂J , j_{1}≠j_{2}} ((π_{j_{1}}) ∪ (π_{j_{2}})) .

Moreover, for each j ∈ J,

inf(H^{1}(G_{F_{0}}, 𝔽_{p})) ≅ inf(H^{1}(G_{F_{0}}, 𝔽_{p})) ∪ (π_{j})

and for each j_{1}, j_{2} ∈ J such that j_{1} ≠ j_{2}, we have (π_{j_{1}}) ∪ (π_{j_{2}}) ≠ 0.

Note that in the last summand of Lemma 4 the sum ranges over subsets {j_{1}, j_{2}}, j_{1} ≠ j_{2} of J and a choice between (π_{j_{1}}) ∪ (π_{j_{2}}) and (π_{j_{2}}) ∪ (π_{j_{1}}) is arbitrary but fixed.

2.3. Residue fields F_{0} with prescribed absolute Galois group.

In our construction of F we choose a residue field F_{0} depending on Υ and p. If Υ = 1 we will simply put F_{0} = ℂ, but when Υ = 0 we require some special properties of F_{0}. In particular, in order that our cyclic extension K = F (\sqrt[p]{a}) have the desired invariant Υ = 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 ξ_{p} ∉ N_{K/F} (K^{×}) by [A, Theorem 3].

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 |F_{0}^{×}/F_{0}^{×p}| is small. As it turns out, we may choose some suitable algebraic infinite extension of ℚ. Finitely generated prop-absolute Galois groups over ℚ and more generally any global field, were nicely classified in [E2]. (See also [E1] and [JP] for related results and techniques.)

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

Lemma 5. [E2, page 84] For each prime p > 2 there exists an algebraic extension F_{0,p} of ℚ such that

G_{F_{0,p}} = ⟨σ, τ | στσ^{−1} = τ^{p+1}⟩_{pro-p}

where the presentation is in the category of pro-p-groups.

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

G_{F_{0,p}}^{ab} ≅ ⟨σ̄, τ̄ | τ̄^{p} = 1⟩ = ℤ_{p} × ℤ/pℤ

for all primes p > 2.

2.4. 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 pth roots of unity in a field F, based only on the structure of G_{F}.

Lemma 6. Suppose that p > 2 and that F is a field with char(F) ≠ p and G_{F} pro-p. Then ξ_{p} ∈ F^{×}.

Proof. Because char(F) ≠ p, there exists a primitive pth root ξ_{p} of unity in F^{s}. If ξ_{p} ∈ F^{s} \ F then F (ξ_{p})/F is a nontrivial Galois extension of degree [F (ξ_{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 ξ_{p} ∈ F^{×} as asserted. □

Now suppose that G_{F} = G_{F_{1}} ⋆ 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 [N, (4.3) Satz] we see that the restriction homomorphism

res : H^{1}(G_{F} , 𝔽_{p}) ⟶ H^{1}(G_{F_{1}}, 𝔽_{p}) ⊕ H^{1}(G_{F_{2}}, 𝔽_{p})  (1)

is an isomorphism. Now given (f)_{F} in H^{1}(G_{F} , 𝔽_{p}), we denote the image res(f)_{F} by

res(f)_{F} = (f)_{G_{F_{1}}} ⊕ (f)_{G_{F_{2}}}.

This notation distinguishes, then, between (f)_{F_{1}}, which denotes φ_{F_{1}}(f) for f ∈ F_{1}^{×}, and (f)_{G_{F_{1}}}, which denotes the projection of res φ_{F} (f) onto the first summand.

One way of interpreting this restriction map is with the following

Lemma 7. Let G_{F} = G_{F_{1}} ⋆G_{F_{2}} be pro-p absolute Galois groups of fields containing a pth root of unity, and suppose that we have the following sequence:

\xymatrix{G_{F} \ar@{->>}[r]^{can} & G_{F_{1}} \ar@{->>}[r] & ℤ/pℤ.}

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}(\sqrt[p]{a_{1}}) and K = F (\sqrt[p]{a}), respectively, where (a)_{G_{F_{1}}} = (a_{1})_{F_{1}} and (a)_{G_{F_{2}}} = 0.

Proof. The surjections are continuous homomorphisms, hence elements of H^{1}(G_{F_{1}}, 𝔽_{p}) and H^{1}(G_{F} , 𝔽_{p}), respectively, and the right-hand surjection is clearly the restriction of the composed surjection. The remainder follows by Kummer theory. □

Lemma 8. Let G_{F} = G_{F_{1}} ⋆ 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

(a)_{F} ∪ (b)_{F} = 0 ∈ H^{2}(G_{F} , 𝔽_{p}).

The lemma follows from [N, (4.1) Satz] and from [Ris, Prop. 7.3, page 191]. 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.

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} , 𝔽_{p}).

Now let H_{p^{3}} be the Heisenberg group of order p^{3}:

H_{p^{3}} = ⟨v_{1}, v_{2}, w | v_{1}^{p} = v_{2}^{p} = w^{p} = 1, v_{2}v_{1} = wv_{1}v_{2},
[v_{1}, w] = [v_{2}, w] = 1⟩

In the case p = 2, H_{8} is the familiar dihedral group D_{4}.

By [M, Corollary, page 523 and Theorem 3(A)], if (a) and (b) are linearly independent, then (a) ∪ (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 (\sqrt[p]{a}, \sqrt[p]{b}) in such a way that

H_{p^{3}}/⟨v_{1}, w⟩ = Gal(F (\sqrt[p]{a})/F) and H_{p^{3}}/⟨v_{2}, w⟩ = Gal(F (\sqrt[p]{b})/F).

Now consider the commutative diagram

\xymatrix{ &  & G_{F} \ar@{->>}[dl]_{δ_{1}} \ar@{->>}[dd]^{β} \ar@{->>}[dr]^{δ_{2}} \\  & G_{F_{1}} \ar@{->>}[dl]_{α_{1}} &  & G_{F_{2}} \ar@{->>}[dr]^{α_{2}} \\ ℤ/pℤ \ar@{^{(}->}[rr]^{1↦v_{1}} &  & H_{p^{3}} \ar[rr]_{⟨v_{1},w⟩↦0; v_{2}↦1} \ar[ll]^{⟨v_{2},w⟩↦0; v_{1}↦1} &  & ℤ/pℤ \ar@{^{(}->}[ll]_{1↦v_{2}}}

Let a_{2} ∈ F_{2}^{×} and b_{1} ∈ F_{1}^{×} satisfy (a_{2})_{F_{2}} = (a)_{G_{F_{2}}} and (b_{1})_{F_{1}} = (b)_{G_{F_{1}}}. Then set K_{1} = F_{1}(\sqrt[p]{b_{1}}) and K_{2} = F_{2}(\sqrt[p]{a_{2}}); these are ℤ/pℤ-extensions of F_{1} and F_{2}, respectively. We may then identify the left-hand ℤ/pℤ in the diagram with Gal(K_{1}/F_{1}) so that α_{1} is the surjection of Galois theory. Similarly, the right-hand ℤ/pℤ may be identified with Gal(K_{2}/F_{2}) with α_{2} the surjection of Galois theory. Finally, the topmost surjections δ_{1} and δ_{2} are canonical.

By Lemma 7, the surjections δ_{1}α_{1} and δ_{2}α_{2} correspond to fields F (\sqrt[p]{b}) and F (\sqrt[p]{a}), respectively. Now because G_{F} = G_{F_{1}} ⋆ G_{F_{2}}, there exists a homomorphism β : G_{F} → H_{p^{3}}, and because v_{1} and v_{2} generate H_{p^{3}}, β 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/pℤ, which corresponds to F (\sqrt[p]{b}), and H_{1}/H is ⟨v_{2}, w⟩. 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/pℤ, which corresponds to F (\sqrt[p]{a}), and H_{2}/H is ⟨v_{1}, w⟩.

Hence (a) ∪ (b) = 0. □

Finally, we close with with a companion to Lemma 7. In Lemma 9 below, π denotes the canonical homomorphism of G to G_{1} which is an identity on G_{1}, and is trivial on G_{2}.

Lemma 9. Let G = G_{1} ⋆ G_{2} be a free product of G_{1} and G_{2} in the category of pro-p-groups. Suppose that A ≅ ℤ/pℤ is a factor group of G_{1} such that the surjection G_{1} → ℤ/pℤ does not factor through ℤ/p^{2}ℤ. Then the following commutative diagram cannot occur:

\xymatrix{G \ar@{->>}[r]^{π} \ar@{->>}[d] & G_{1} \ar@{->>}[d] \\ ℤ/p^{2}ℤ \ar@{->>}[r] & A.}

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]^{π^{ab}} \ar@{->>}[d]_{α} & G_{1}^{ab} \ar@{->>}[d]^{γ} \\ ℤ/p^{2}ℤ \ar@{->>}[r]^{β} & A.}

But G^{ab} ≅ G_{1}^{ab} × G_{2}^{ab} and the canonical surjection onto G_{1}^{ab} is given by the projection map. Let δ be a splitting map of the projection map. Then γ = βαδ, contradicting the hypothesis. □

3. Proof of the Theorem

First we define fields F_{0}, F_{1}, F_{2}, and F using our given cardinal numbers d, e, and Υ, 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 Υ.

3.1. Constructing F_{0}, F_{1}, F_{2}, and F.

If Υ = 1 then let F_{0} = ℂ. If Υ = 0 and p = 2, let F_{0} = ℝ. (See Proposition 1 for alternative choices in these two cases.) If Υ = 0 and p > 2 then let F_{0,p} be the algebraic extension of ℚ of Lemma 5. In the first two cases we see trivially that ξ_{p} ∈ F_{0}, and in the last case ξ_{p} ∈ F_{0} by Lemma 6. Observe that

dim_{𝔽_{p}} F_{0}^{×}/F_{0}^{×p} = \begin{cases}0, if Υ = 1; \\ 1, if Υ = 0, p = 2; \\ 2, if Υ = 0, p > 2. \end{cases}  (2)

We next construct the field F_{1}. Because Υ = 0 implies 1 ≤ d, for either choice of Υ ∈ {0, 1} there exists a well-ordered set I_{1} such that |I_{1}| + 1 = d + 2Υ. Let Γ_{1} = ℤ_{(p)}^{(I_{1})} be a direct sum of |I_{1}| copies of ℤ_{(p)}. Then Γ_{1} is a linearly ordered abelian group. Finally set F_{1} := F_{0}((Γ_{1})). From Lemmas 2 and 3 it follows that G_{F_{1}} is a pro-p-group and

dim_{𝔽_{p}} F_{1}^{×}/F_{1}^{×p} = dim_{𝔽_{p}} F_{0}^{×}/F_{0}^{×p} + |I_{1}|.

Hence

dim_{𝔽_{p}} F_{1}^{×}/F_{1}^{×p} = \begin{cases}d + 1, & if Υ = 1; \\ d, & if Υ = 0, p = 2; \\ d + 1, & if Υ = 0, p > 2. \end{cases}  (3)

Similarly, we construct F_{2} as follows. Because p > 2 and also p = 2 and Υ = 1 implies e > 0, there exists a well-ordered set I_{2} such that 1 + |I_{2}| = e in either of the cases p > 2 or Υ = 1, p = 2, and |I_{2}| = e in the case Υ = 0, p = 2. Then again Γ_{2} = ℤ_{(p)}^{(I_{2})} is a linearly ordered abelian group. We set F_{2} := ℂ((Γ_{2})). Then from [K, pages 3 and 4] it follows that G_{F_{2}} ≅ ℤ_{p}^{I_{2}}, the topological product of |I_{2}| copies of ℤ_{p}. In particular

e = \begin{cases}dim_{𝔽_{p}} F_{2}^{×}/F_{2}^{×p} + 1, & if p > 2 or p = 2 and Υ = 1; \\ dim_{𝔽_{p}} F_{2}^{×}/F_{2}^{×p}, & if p = 2, Υ = 0. \end{cases} (4)

From Theorem 2 we see that there exists a field F of characteristic zero, such that G_{F} = G_{F_{1}} ⋆ 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 6 we see that F contains a primitive pth root of unity.

3.2. Constructing K/F.

We define the cyclic extension K/F of degree p as K = F (\sqrt[p]{a}) where [a] ∈ F^{×}/F^{×p} is chosen via the isomorphism (1).

If Υ = 1 then let (a_{1})_{F_{1}} be any nontrivial element in H^{1}(G_{F_{1}}, 𝔽_{p}), which is possible since 1 ≤ dim_{𝔽_{p}} H^{1}(G_{F_{1}}, 𝔽_{p}) = d + 1. Let (a_{2})_{F_{2}} = 0 in H^{2}(G_{F_{2}}, 𝔽_{p}).

Now suppose Υ = 0 and p > 2. We denote the fixed field of the factor ℤ_{p} in G_{F_{0}}^{ab} ≅ ℤ_{p} × ℤ/pℤ acting on a maximal abelian extension F_{0}^{ab} of F_{0} as K_{0} := F_{0}(\sqrt[p]{b}). (See the discussion following Lemma 5.) Because F_{0}^{×}/F_{0}^{×p} is naturally isomorphic to a subgroup of F_{1}^{×}/F_{1}^{×p}, we may set (a_{1})_{F_{1}} = inf(b)_{F_{0}} ≠ 0 in H^{1}(G_{F_{1}}, 𝔽_{p}) and (a_{2})_{F_{2}} = 0 in H^{2}(G_{F_{2}}, 𝔽_{p}).

Finally assume that Υ = 0 and p = 2. We set (a_{1})_{F_{1}} = (−1)_{F_{1}} ≠ 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 (1) there exists a ∈ F^{×} such that (a)_{F} ≠ 0 and (a)_{G_{F_{i}}} = (a_{i})_{F_{i}}, i = 1, 2. Since we have already determined that ξ_{p} ∈ F^{×}, we see that K = F (\sqrt[p]{a}) is a cyclic extension of degree p. Hence it remains to show that the arithmetic invariants d(K/F), e(K/F), and Υ(K/F) coincide with the prescribed cardinal numbers d, e, and Υ respectively.

3.3. 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} , 𝔽_{p}). Recall that Lemma 8 tells us that if c_{1}, c_{2} ∈ F^{×} with (c_{1})_{G_{F_{1}}} = (c_{2})_{G_{F_{2}}} = 0 then

(c_{1})_{F} ∪ (c_{2})_{F} = 0 ∈ H^{2}(G_{F} , 𝔽_{p}).  (5)

Moreover, recall that by [N, Satz 4.1]

H^{2}(G_{F} , 𝔽_{p}) ≅ H^{2}(G_{F_{1}}, 𝔽_{p}) × H^{2}(G_{F_{2}}, 𝔽_{p}),

where the isomorphism is induced by the restriction maps

res_{i} : H^{2}(G_{F} , 𝔽_{p}) → H^{2}(G_{F_{i}}, 𝔽_{p}), i = 1, 2.

Because these restriction maps commute with cup product maps ([Ris, Proposition 7.3, page 191]), we see that if c_{1}, c_{2} ∈ F^{×} such that (c_{1})_{G_{F_{2}}} = (c_{2})_{G_{F_{2}}} = 0 in H^{2}(G_{F_{2}}, 𝔽_{p}) then

(c_{1})_{F} ∪ (c_{2})_{F} ≠ 0 iff (c_{1})_{G_{F_{1}}} ∪ (c_{2})_{G_{F_{1}}} ≠ 0,  (6)

where the first cup product lies in H^{1}(G_{F} , 𝔽_{p}) and the second cup product lies in H^{1}(G_{F_{1}}, 𝔽_{p}). 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 ϕ ∈ H^{1}(G_{F} , 𝔽_{p}):

ann_{F_{1}} ϕ := {η ∈ H^{1}(G_{F} , 𝔽_{p}) | η ∪ ϕ = 0}.

For x ∈ F^{×} we have x ∈ N_{K/F} (K^{×}) if and only if (x)_{F} ∪ (a)_{F} = 0 in H^{2}(G_{F} , 𝔽_{p}). Therefore, using (5) and (6) with the fact that in every case (a)_{G_{F_{2}}} = 0,

e(K/F) = dim_{𝔽_{p}} N(K^{×})/F^{×p}
= dim_{𝔽_{p}}{(x)_{F} ∈ H^{1}(G_{F} , 𝔽_{p}) | (a)_{F} ∪ (x)_{F} = 0}
= dim_{𝔽_{p}}{(y)_{F_{1}} ∈ H^{1}(G_{F_{1}}, 𝔽_{p}) | (a)_{G_{F_{1}}} ∪ (y)_{F_{1}} = 0}
+ dim_{𝔽_{p}} H^{1}(G_{F_{2}}, 𝔽_{p})
= dim_{𝔽_{p}} ann_{F_{1}}(a)_{G_{F_{1}}} + dim_{𝔽_{p}} F_{2}^{×}/F_{2}^{×p}.

If Υ = 1 then H^{1}(G_{F_{0}}, 𝔽_{p}) = 0, and so by Lemma 4 we have dim_{𝔽_{p}} ann_{F_{1}}(a)_{G_{F_{1}}} = 1.

If p > 2 and Υ = 0 then by Lemma 4,

dim_{𝔽_{p}} ann_{F_{1}}(a)_{G_{F_{1}}} = dim_{𝔽_{p}} ann_{F_{0}}(b)_{F_{0}},

where b was chosen so that K_{0} = F_{0}(\sqrt[p]{b}) was the fixed field of the factor ℤ_{p} in G_{F_{0}}^{ab}. Now because (b)_{F_{0}} ∪ (b)_{F_{0}} = 0 as an identity in H^{2}(G_{F} , 𝔽_{p}) for p > 2,

1 ≤ dim_{𝔽_{p}} ann_{F_{0}}(b)_{F_{0}} ≤ 2.

On the other hand, let c ∈ F^{×} \ F^{×p} be defined so that F_{0}(\sqrt[p]{c}) is contained in the fixed field of the factor ℤ/pℤ in G_{F_{0}}^{ab}. Then by (2), {(b)_{F_{0}}, (c)_{F_{0}}} spans H^{1}(G_{F_{0}}, 𝔽_{p}). Furthermore, since by Lemma 5, H_{p^{3}} is not a quotient of G_{F_{0,p}}, (b)_{F_{0}} ∪ (c)_{F_{0}} ≠ 0 by [M, Corollary, page 523 and Theorem 3(A)]. We conclude that dim_{𝔽_{p}} ann_{F_{0}}(b)_{F_{0}} = 1.

Finally, if p = 2 and Υ = 0 then again by Lemma 4

dim_{𝔽_{2}} ann_{F_{1}}(a)_{G_{F_{1}}} = dim_{𝔽_{2}} ann_{F_{0}}(−1)_{F_{0}}.

As F_{0} = ℝ, dim_{𝔽_{2}} F_{0}^{×}/F_{0}^{×2} = 1 and (−1)_{ℝ} ∪ (−1)_{ℝ} ≠ 0, yielding dim_{𝔽_{2}} ann_{F_{1}}(a)_{G_{F_{1}}} = 0.

Combining our results with (4), we have that e(K/F) = e.

Now we turn to a similar calculation of d(K/F). Again using (5) and (6) with the fact that in every case (a)_{G_{F_{2}}} = 0,

d(K/F) = dim_{𝔽_{p}}(H^{1}(G_{F} , 𝔽_{p})/ ann_{F} (a)_{F})
= dim_{𝔽_{p}}(H^{1}(G_{F_{1}}, 𝔽_{p})/ ann_{F_{1}}(a)_{G_{F_{1}}})
= d.

For this last equality we use (3) together with calculations of the dimension of ann_{F_{1}}(a)_{G_{F_{1}}} already achieved. Hence d(K/F) = d in all cases.

3.4. Determining Υ via quotients of G_{F}.

It remains to show that Υ(K/F) = Υ. First consider the case Υ = 1. Since F_{0} = ℂ, ξ_{p} is a pth power in F_{1}, and F_{1}(\sqrt[p]{a}) embeds in a ℤ/p^{2}ℤextension F_{1}( \sqrt[p^{2}]{a}) of F_{1}. Then the surjection G_{F_{1}} → Gal(K_{1}/F_{1}) factors through ℤ/p^{2}ℤ. Following the surjection with the canonical surjection G_{F} → G_{F_{1}}, we see that ℤ/p^{2}ℤ is a factor group of G_{F}. Moreover, by Lemma 7, the surjection G_{F} → ℤ/pℤ corresponds to K. Hence K/F embeds in a ℤ/p^{2}ℤ-extension of F. By [A, Theorem 3], ξ_{p} ∈ N_{K/F} (K^{×}). Therefore Υ(K/F) = 1 = Υ.

Now consider the case Υ = 0 and p > 2. Because K_{0} = F_{0}(\sqrt[p]{b}) does not embed in a ℤ/p^{2}ℤ-extension of F_{0}, ξ_{p} ∉ N_{K_{0}/F_{0}}(K_{0}^{×}). (See 3.2 for the definition of K_{0}.) Hence (b)_{F_{0}} ∪ (ξ_{p})_{F_{0}} ≠ 0 in H^{1}(G_{F_{0}}, 𝔽_{p}), and, by Lemma 4, (inf(b)) ∪ (ξ_{p})_{F_{1}} ≠ 0 in H^{1}(G_{F_{1}}, 𝔽_{p}) as well. Choose b_{1} ∈ F_{1}^{×} so that (b_{1})_{F_{1}} = inf(b)_{F_{0}} and set K_{1} = F_{1}(\sqrt[p]{b_{1}}). Then ξ_{p} ∉ N_{K_{1}/F_{1}}(K_{1}^{×}) and by [A, Theorem 3] 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}} → Gal(F_{1}(\sqrt[p]{b})/F_{1}) does not factor through ℤ/p^{2}ℤ. From Lemmas 7 and 9 we see that surjection G_{F} → Gal(K/F) does not factor through ℤ/p^{2}ℤ. Again by [A, Theorem 3] we conclude that ξ_{p} ∉ N_{K/F} (K^{×}) and therefore Υ(K/F) = 0 = Υ.

Finally consider the case Υ = 0 and p = 2. Because −1 ∉ N_{ℂ/ℝ}(ℂ) from Lemma 4 we conclude that −1 ∉ N_{K_{1}/F_{1}}(K_{1}^{×}). By [A, Theorem 3] the surjection G_{F_{1}} → Gal(K_{1}/F_{1}) does not factor through ℤ/4ℤ. As in the previous case, by Lemmas 7 and 9 we see that K/F does not embed in a ℤ/4ℤ-extension of F. Again by [A, Theorem 3] we see that −1 ∉ N_{K/F} (K^{×}) and therefore Υ(K/F) = 0 = Υ.

Hence we have checked in all cases that Υ(K/F) = Υ, and our proof of Theorem 1 is now complete. □

Remark. In [EH, Lemma 1.2 and Proposition 1.3] 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 Υ = 0 in our proof above, yield the following proposition.

Proposition 1. Suppose that p is a prime number, and let d, e, and Υ ∈ {0, 1} be cardinal numbers such that if Υ = 0 then 1 ≤ d, if p > 2 then 1 ≤ e, and if p = 2 and Υ = 1 then 1 ≤ e. Then there exists a field F containing ℚ(ξ_{p}) and a cyclic Galois extension K of degree p over F such that

e(K/F) = e, d(K/F) = d, and Υ(K/F) = Υ.

Moreover

tr. deg(F/ℚ) ≤ 1 + max{e, d + 1}.

In particular if d, e ∈ ℕ ∪ {0}, there exists a Galois cyclic extension K/F of degree p with prescribed invariants d, e, and Υ of finite transcendence degree over its prime field ℚ.

4. Acknowledgements

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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Department of Mathematics, Middlesex College, University of Western Ontario, London, Ontario N6A 5B7 CANADA

E-mail address: minac@uwo.ca

Department of Mathematics, Davidson College, Box 7046, Davidson, North Carolina 28035-7046 USA

E-mail address: joswallow@davidson.edu