CONDITIONS SATISFIED BY CHARACTERISTIC POLYNOMIALS IN FIELDS AND DIVISION ALGEBRAS

Z. REICHSTEIN AND B. YOUSSIN

Abstract. Suppose E/F is a field extension. We ask whether or not there exists an element of E whose characteristic polynomial has one or more zero coefficients in specified positions. We show that the answer is frequently “no”. We also prove similar results for division algebras and show that the universal division algebra of degree n does not have an element of trace 0 and norm 1.

Contents

1. Introduction 2 Notational conventions 3 Main results 3 Acknowledgements 4 2. The Going Down Theorem and its applications 4 S_{n}-varieties 4 PGL_{n}-varieties 5 3. Abelian subgroups 6 Abelian subgroups of S_{n} 6 Abelian subgroups of PGL_{n} 7 4. Proof of Theorem 1.5 8 The group H 8 The variety X 8 The variety Y 10 Conclusion of the proof 10 Refinements 11 5. Proof of Theorem 1.7 11 6. Systems of the form σ^{(m_{1})}(x) = σ^{(m_{2})}(x) = 0 13 7. A further generalization 15 8. Equations in octonion algebras 18 Preliminaries 18 G_{2}-equivariant maps 18 G_{2}-invariant polynomials 19 A system of equations 19 References 20

1991 Mathematics Subject Classification. 12E05, 12E12, 12E15, 14L30, 16A39.

Z. Reichstein was partially supported by NSF grant DMS-9801675.

1. Introduction

Let E/F be a field extension of degree n and det : E ⟶ F be the norm function. For x ∈ E, we define σ^{(i)}(x) by

det(λ1_{F} − x) = λ^{n} + σ^{(1)}(x)λ^{n−1} + ⋯ + σ^{(n−1)}(x)λ + σ^{(n)}(x) . (1.1)

In particular, σ^{(1)}(x) = − tr(x) and σ^{(n)}(x) = (−1)^{n} det(x). In the sequel, whenever we write σ^{(i)}(x), we shall always understand i to be an integer between 1 and n. If the reference to the extension E/F is not clear from the context, we will sometimes write σ_{E/F}^{(i)} (x) in place of σ^{(i)}(x) ∈ F.

If A is a central simple algebra of degree n with center F then we can define σ^{(i)} = σ_{A/F}^{(i)} in the same way. Here det in formula (1.1) should be intepreted as the reduced norm in A ⊗_{F} F (λ).

A number of interesting results, both in the theory of polynomials and in the theory of central simple algebras, can be stated in terms of the existence (or nonexistence) of nontrivial solutions to systems of equations of the form

σ^{(i)}(x) = 0 for i = i_{1}, …, i_{r} .  (1.2)

Example 1.1. (Hermite [H], Joubert [J]; see also Coray [C]) If E/F is a field extension of degree 5 or 6 and char(F) ≠ 3 then there exists an element x ∈ E such that E = F (x) and σ^{(1)}(x) = σ^{(3)}(x) = 0.

In classical language, this means that for n = 5 or 6 every polynomial f(t) = t^{n} + a_{1}t^{n−1} + ⋯ + a_{n} ∈ F [t] can be reduced, via the Tschirnhaus transformation t ↦ x, to the form f(t) = t^{n} + b_{1}t^{n−1} + ⋯ + b_{n} ∈ F [t] with b_{1} = b_{3} = 0; for details we refer the reader to [BR].

Example 1.2. Let A be a central simple algebra of degree n whose center contains a primitive nth root of unity. Then A is cyclic iff there exists an element x such that

σ^{(1)}(x) = ⋯ = σ^{(n−1)}(x) = 0 .

A conjecture of Albert asserts that every A of prime (or, equivalently, square-free) degree is cyclic. This conjecture is known to be true for n = 2, 3 and 6 (see [Ro_{1}, Section 3.2]); the remaining cases are open.

Example 1.3. (Haile [Ha]; see also Brauer [Ro_{3}, Proposition 7.1.43]) Suppose A is a central simple algebra of degree n with center F. Then there exists an (n − 1)- dimensional F -subspace W of A such that σ^{(1)}(x) = σ^{(n−1)}(x) = 0 for any x ∈ W.

Example 1.4. (Rowen [Ro_{2}, Corollary 5]) If A is a central simple algebra of odd degree with center F then there exists an element x ∈ A − {0} such that σ^{(1)}(x) = σ^{(2)}(x) = 0.

Note that if char(F) ≠ 2, this follows easily from a theorem of Springer (see e.g., [Re_{1}, Remark 14.3]); however, the above result is true even if char(F) = 2.

In [Re_{1}] the first author showed that in many cases equations of the form σ^{(i)}(x) = 0 or tr(x^{i}) = 0 and systems of the form σ^{(1)}(x) = σ^{(i)}(x) = 0 or tr(x) = tr(x^{i}) = 0 do not have nontrivial solutions. In particular, the theorem of Hermite and Joubert, cited in Example 1.1, fails for field extensions of degree n = 3^{m} or 3^{m} + 3^{l}, with m > l ≥ 0. In this paper we revisit this subject from a more geometric point of view.

Notational conventions. Throughout this paper n will denote the degree of the field extension or division algebra we are considering, and sqf(n) will denote the square-free part of n. We will always work over a fixed ground field k.

Let K be a field containing a primitive rth root of unity ζ_{r} (in particular, we assume that r is prime to char(K)), and let z, w ∈ K. Recall that a symbol algebra (z, w)_{r} is defined as

(z, w)_{r} = K{x, y}/(x^{r} = z , y^{r} = w , yx = ζ_{r}xy) ;  (1.3)

cf. [Ro_{3}, p. 194]. We now define the algebra D_{n} as follows. Write n = p_{1} … p_{s} as a product of (not necessarily distinct) primes. Let K = k(z_{1}, w_{1}, …, z_{s}, w_{s}), where z_{1}, w_{1}, …, z_{s}, w_{s} are independent variables over k and let

D_{n} = (z_{1}, w_{1})_{p_{1}} ⊗_{K} ⋯ ⊗_{K} (z_{s}, w_{s})_{p_{s}} .  (1.4)

Note that D_{n} is a division algebra of degree n and exponent sqf(n), with center K.

Finally recall that the universal division algebra UD(n) is the subalgebra of M_{n}(k(s_{ij}, t_{ij})) generated, as a division algebra, by two generic n × n-matrices (s_{ij}) and (t_{ij}). Here s_{ij} and t_{ij} are 2n^{2} independent variables over k. For details of this construction, see, e.g., [Ro_{1}, Section 3.2].

Main results.

Theorem 1.5. Suppose char(k) |̸ n! and D = D_{n} or UD(n). Then the system

\{ \begin{matrix}σ^{(i)}(x_{1}) = ⋯ = σ^{(i)}(x_{m}) \\ σ^{(j)}(x_{1} … x_{m}) = 0 \end{matrix}  (1.5)

has no nontrivial solutions in D, provided that i and m are divisible by sqf(n).

Here, as usual, a solution (x_{1}, …, x_{s}) is trivial if x_{1} = ⋯ = x_{s} = 0 and nontrivial otherwise. Note that the assertion of the theorem for UD(n) is a formal consequence of the assertion for D_{n}, because of the specialization property of UD(n). However, our proof will treat the two cases in parallel, since both are proved by the same argument. Theorem 1.5 can be generalized in several directions; some generalizations are discussed at the end of Section 4.

We now record three consequences of Theorem 1.5, which we feel deserve a special mention.

Corollary 1.6. Suppose char(k) |̸ n!, D = D_{n} or UD(n), and m is divisible by sqf(n).

(a) σ^{(m)}(x) ≠ 0 for any x ∈ D − {0}.

(b) If det(x_{1}) = ⋯ = det(x_{m}) for some x_{1}, …, x_{m} ∈ D − {0} then tr(x_{1} … x_{m}) ≠ 0.

(c) D does not have an element of (reduced) norm 1 and (reduced) trace 0.

To prove part (a), we assume the contrary and substitute i = m, x_{1} = x and x_{2} = ⋯ = x_{m} = 0 into (1.5) to obtain a contradiction. To prove part (b), we apply Theorem 1.5 with i = n and j = 1. Finally, if det(x) = 1 then setting x_{1} = x and x_{2} = ⋯ = x_{m} = 1 in part (b), we obtain tr(x) ≠ 0, thus proving part (c).

The commutative counterpart of the universal division algebra is the general field extension L_{n}/K_{n} defined as follows:

K_{n} = k(a_{1}, …, a_{n}) and L_{n} = K_{n}[x]/(x^{n} + a_{1}x^{n−1} + ⋯ + a_{n}) ,  (1.6)

where a_{1}, …, a_{n} are algebraically independent indeterminates over k.

Theorem 1.7. Let n_{1} and n_{2} be positive integers, and L_{n}/K_{n} be the general field extension of degree n = n_{1} + n_{2}. Then the system of equations

tr(x^{m_{1}}) = tr(x^{m_{2}}) = 0  (1.7)

has no nontrivial solutions x ∈ L_{n}^{∗}, provided that (i) n_{1}n_{2} ≠ 0 and (−\frac{n_{2}}{n_{1}} )^{m_{2}−m_{1}} ≠ 1 in k. (ii) each sqf(n_{i}) (i = 1, 2) divides m_{1} or m_{2} (and possibly both).

Note that if char(k) = 0 then condition (i) holds unless m_{1} = m_{2} or n_{1} = n_{2} and m_{2} − m_{1} is even. If we replace (i) by a more complicated condition, we can also show that the system σ^{(m_{1})}(x) = σ^{(m_{2})}(x) = 0 has no nontrivial solutions; see Section 6.

It is interesting to note that Theorem 1.5 and Corollary 1.6 remain true if D is replaced by L_{n}; see Remark 4.6. On the other hand, Theorem 1.7 fails if L_{n} is replaced by UD(n); see Remark 5.2.

All of the main results in this paper are proved by the same general method. based on the Going Down Theorem 2.1. This method is outlined in Section 2. In particular, our proofs of Theorems 1.5 and 1.7, given in Sections 4 and 5, are applications of Propositions 2.4 and 2.2 respectively. Proposition 2.4 says a system of equations, such as (1.5), has no nontrivial solutions in a “sufficiently generic” division algebra if a certain projective PGL_{n}-variety, constructed from this system, does not have H-fixed points for some abelian subgroup H of PGL_{n}. Proposition 2.2 gives a similar criterion for nonexistence of solutions in field extensions. Other applications of this approach and some generalizations are presented in Sections 6– 8.

Acknowledgements. The authors would like to thank A. R. Wadsworth for helpful discussions.

2. The Going Down Theorem and its applications

The following result will play a key role in the sequel. A simple proof, due to Kollár and Szabó, can be found in [RY_{1}, Appendix]. Assume that k is an algebraically closed base field, and that all varieties, group actions and maps are defined over k.

Theorem 2.1 (The Going Down Theorem). Let H be a finite abelian group acting on algebraic varieties X and Y and let f: X −→ Y be an H-equivariant rational map. If X has a smooth H-fixed point and Y is projective then Y has an H-fixed point.

S_{n}-varieties. Let L/K be a separable field extension of degree n, let L′ be the normal closure of L over K, and Gal(L′/K) = G. Note that G acts on the set of embeddings L ↪ L′ and thus is naturally realized as a transitive subgroup of S_{n}. For each i = 1, …, n choose g_{i} ∈ S_{n} such that g_{i}(1) = i. The embedding of G in S_{n} defines a (permutation) action of G on 𝔸^{n} and thus a diagonal actions on (𝔸^{n})^{m} for every m ≥ 1.

Let P (x_{11}, …, x_{1n}; …; x_{m1}, …, x_{mn}) ∈ k[(𝔸^{n})^{m}] be a G-invariant polynomial and let a_{1}, …, a_{m} ∈ L. Then we can define P (a_{1}, …, a_{m}) as P (a_{11}, …, a_{1n}; …; a_{m1}, …, a_{mn}), where a_{ij} = g_{j}(a_{i}) ∈ L′. A priori, P (a_{1}, …, a_{m}) ∈ L′; however, since P is G-invariant polynomial, P (a_{1}, …, a_{m}) actually lies in (L′)^{G} = K.

In the sequel we shall assume that K is finitely generated over k (and hence, so are L and L′).

Proposition 2.2. Let Y be the subvariety of ℙ((𝔸^{n})^{m}) given by G-invariant homogeneous polynomial equations P_{1} = ⋯ = P_{s} = 0. Suppose that Y does not have H-fixed points for some abelian subgroup H ⊂ G. Assume that there exists a G-variety X which has a smooth H-fixed point and such that k(X) = L′ as fields with G-action. Then the system of equations

P_{1}(a_{1}, …, a_{m}) = ⋯ = P_{s}(a_{1}, …, a_{m}) = 0  (2.1)

has no nontrivial solutions in L.

We remark that if char(k) = 0 then a G-variety X such that k(X) = L′ (as G-fields) always exists; see [Re_{2}, Proposition 8.6 and Example 8.4c]. Moreover, we can choose X to be smooth and projective; see [RY_{2}, Proposition 2.2]. In view of Theorem 2.1, the presence of an H-fixed point on such an X is a birational invariant, i.e., is independent of the choice of the (smooth projective) model.

Proof. Suppose (a_{1}, …, a_{m}) ∈ L^{m} ⊂ k(X)^{m} is a non-trivial solution of (2.1) and let a_{i1}, …, a_{in} be the conjugates of a_{i} in L′. Then

f: x ↦ [a_{11}(x) : a_{12}(x) : ⋯: a_{mn}(x)]

is a G-equivariant rational map X −→ ℙ((𝔸^{n})^{m}). By our choice of a_{1}, …, a_{m}, the image of f lies in Y. Applying Theorem 2.1 to the rational map f: X −→ Y, we conclude that Y has an H-fixed point, a contradiction.

In the sequel we shall use use Proposition 2.2 only for m = 1; the statement for general m is intended to make it parallel to Proposition 2.4 below.

PGL_{n}-varieties. Let P ∈ k[(M_{n})^{m}]^{PGL^{n}}; it is a polynomial in the entries of m matrices U_{1}, …, U_{m} invariant under simultaneous conjugation. If A is a central simple algebra of degree n and a_{1}, …, a_{m} ∈ A then we can define P (a_{1}, …, a_{m}) as follows. Split A by the algebraic closure \overline{K} of K: A ⊗_{K} \overline{K} ≃ M_{n}(\overline{K}). Thus A ↪ M_{n}(\overline{K}), and we can evaluate P (a_{1}, …, a_{m}) ∈ \overline{K}.

Lemma 2.3. P (a_{1}, …, a_{m}) lies in K and is independent of the choice of the isomorphism A ⊗_{K} \overline{K} ≃ M_{n}(\overline{K}).

Proof. Any two choices of the isomorphism A⊗_{K} \overline{K} ≃ M_{n}(\overline{K}) differ by conjugation by some g ∈ PGL_{n}(\overline{K}). Since P is PGL_{n}-invariant, conjugation by g does not change the value of P (a_{1}, ..., a_{m}).

Consider the action of Gal(\overline{K}/K) on M_{n}(\overline{K}); for any σ ∈ Gal(\overline{K}/K) and B_{1}, …, B_{n} ∈ M_{n}(\overline{K}), P (σ(B_{1}), …, σ(B_{n})) = σ(P (B_{1}, …, B_{m})). The composition

M_{n}(\overline{K}) ⟶^{∼} A ⊗_{K} \overline{K} ⟶^{Id ⊗σ^{−1}} A ⊗_{K} \overline{K} ⟶^{∼} M_{n}(\overline{K}) ⟶^{σ} M_{n}(\overline{K})

is an automorphism of M_{n}(\overline{K}) whose restriction to the center \overline{K} is trivial. Hence, this composition is given by conjugation by some g ∈ PGL_{n}(\overline{K}). It follows that for a_{1}, …, a_{m} ∈ A, P (a_{1}, …, a_{m}) is fixed by Gal(\overline{K}/K) and thus lies in K.

Note that the Lemma is an immediate consequence of the fact that k[(M_{n})^{m}]^{PGL^{n}} is generated by elements of the form σ^{(i)}(U), where U is a monomial in the mmatrices U_{1}, …, U_{m}. The latter was proved by Sibirskii [Si] and Procesi [P1] in the case char(k) = 0 and, more recently, by Donkin [D] in prime characteristic. The elementary argument given above allows us to avoid appealing to this more difficult result.

Next we recall that is F be a finitely generated field extension of k then an element of H^{1}(F, PGL_{n}) may be interpreted either as a central simple algebra D of degree n with center F or, alternatively, as a generically free PGL_{n}-variety X such that k(X)^{PGL^{n}} = F. It is shown in [Re_{1}] (under the assumption char(k) = 0) that D =^{∼} RMaps_{PGL_{n}}(X, M_{n}) = the algebra of PGL_{n}-equivariant rational maps from X to M_{n}; see also [RY_{2}, Section 3]. Note that the above isomorphism is an isomorphism of F -algebras, where we identify f ∈ F = k(X)^{PGL^{n}} with the PGL_{n}- equivariant rational map X −→ M_{n}(k) given by x ↦ f(x)I_{n}. (Here I_{n} denotes the n × n-identity matrix.)

Proposition 2.4. Let Y be the subvariety of ℙ((M_{n})^{m}) cut out by PGL_{n}-invariant homogeneous polynomial equations P_{1} = ⋯ = P_{s} = 0. Suppose Y has no fixed points for some finite abelian subgroup H of PGL_{n}. Then the system of equations

P_{1}(x_{1}, …, x_{m}) = ⋯ = P_{s}(x_{1}, …, x_{m}) = 0  (2.2)

has no nontrivial solutions in any central simple algebra D of the form D = RMaps_{PGL_{n}}(X, M_{n}), where X is a generically free PGL_{n}-variety which has a smooth H-fixed point.

Proof. Suppose the system (2.2) has a nontrivial solution (x_{1}, …, x_{m}). As D = RMaps_{PGL_{n}}(X, M_{n}), each x_{i} can be interpreted as a rational PGL_{n}-invariant map X −→ M_{n}; collectively, these elements define a rational PGL_{n}-equivariant map f: X −→ ℙ((M_{n})^{m}). By our choice of x_{1}, …, x_{m}, the image of this map lies in Y. By Theorem 2.1, Y has a H-fixed point, a contradiction.

3. Abelian subgroups

In order to use Propositions 2.2 and 2.4, we need a description of abelian subgroups H of S_{n} and PGL_{n}. In this section we introduce the abelian subgroups that will be used in subsequent applications.

We shall assume that the base field k contains all roots of unity. For a finite abelian group A of order prime to char(k), we shall denote its dual group Hom(A, k^{∗}) by A^{∗}.

Abelian subgroups of S_{n}. Let A = {a_{1}, …, a_{n}} be an abelian group of order n. The right multiplication action of A on itself gives rise to an embedding

ψ_{A}: A ↪ S_{n} .

(Note that if we relabel the elements of A, ψ_{A} will change by an inner automorphism of S_{n}.) Given a character χ: A ⟶ k^{∗}, let

R_{χ} = (χ(a_{1}), …, χ(a_{n})) .

It is easy to see that k^{n} = ⨁_{χ∈A^{∗}} Span_{k}(R_{χ}) is a decomposition of k^{n} as a direct sum of 1-dimensional character spaces for the permutation action of A on k^{n} (via ψ_{A}); moreover, the character associated to Span_{k}(R_{χ}) is precisely χ^{−1}.

In the sequel we will be interested in the permutation action of

H = ψ_{A_{1}}(A_{1}) × ψ_{A_{2}}(A_{2}) ⊂ S_{n_{1}} × S_{n_{2}} ⊂ S_{n}  (3.1)

on k^{n}. Here A_{1} and A_{2} are abelian groups of order n_{1} and n_{2} respectively and n = n_{1} + n_{2}. For future reference, we decompose this action as a direct sum of character spaces. We shall write elements of k^{n} = k^{n_{1}+n_{2}} as (R′, R′′), where

R′ ∈ k^{n_{1}} and R′′ ∈ k^{n_{2}}. Let V_{0} = \{(\underbrace{a, …, a}_{n_{1} times}, \underbrace{b, …, b}_{n_{2} times}) | a, b ∈ k\}.

Lemma 3.1.

k^{n} = V_{0} ⊕ (⨁_{χ∈A_{1}^{∗}} Span_{k}(R_{χ}, 0)) ⊕ (⨁_{η∈A_{2}^{∗}} Span_{k}(0, R_{η}))

is a decomposition of k^{n} as a direct sum of character spaces for the H-action defined above. Here V_{0} is a 2-dimensional subspace with trivial associated character; the remaining n − 2 summands are 1-dimensional subspaces with distinct nontrivial characters.

Proof. The proof of this lemma amounts to verifying that the summands of the above decomposition are, indeed, character spaces and finding their characters. We leave the details of the reader.

Abelian subgroups of PGL_{n}. Let A be an abelian subgroup of order n and V = k[A]. The group A acts on V by the regular representation a ↦ P_{a} ∈ GL(V), where

P_{a}(∑_{b∈A} c_{b}b) = ∑_{b∈A} c_{b}ab

for any a ∈ A and c_{b} ∈ k. The dual group A^{∗} acts on V by the representation χ ↦ D_{χ} ∈ GL(V), where

D_{χ}(∑_{a∈A} c_{a}a) = ∑_{a∈A} c_{a}χ(a)a

for any χ ∈ A^{∗} and c_{a} ∈ k. Note that in the basis {a | a ∈ A} of V, each P_{a} is represented by a permutation matrix and each D_{χ} is represented by a diagonal matrix; this explains our choice of the letters P and D. It is easy to see that

D_{χ}P_{a} = χ(a)P_{a}D_{χ} ;  (3.2)

hence, we have constructed an embedding

ϕ_{A}: A × A^{∗} ↪ PGL(V) = PGL_{n}  (3.3)

given by (a, χ) ↦ \overline{P_{a}} · \overline{D_{χ}}, where \overline{P_{a}} and \overline{D_{χ}} are the elements of PGL(V), represented, respectively, by P_{a} and D_{χ} ∈ GL(V).

For future reference we record two simple lemmas.

Lemma 3.2. For each a ∈ A and χ ∈ A^{∗}, V_{a,χ} = Span_{k}(P_{a}D_{χ}) is a 1-dimensional H-invariant subspace of M_{n}, with associated character (b, η) ↦ χ^{−1}(b)η(a). Moreover, the n^{2} matrices P_{a}D_{χ} form a k-basis of M_{n}.

Proof. The first assertion is immediate from (3.2). Since the n^{2} characters associated to the spaces V_{a,χ} are distinct, the second assertion now follows from linear independence of characters.

Lemma 3.3. Let A be an abelian group of order n and (a, χ) be an element of order c in A × A^{∗}.

(a) (P_{a}D_{χ})^{c} = ϵI_{n}, where ϵ = χ(a)^{\frac{1}{2} c(c−1)} = ±1 and I_{n} is the n × n-identity matrix.

(b) The characteristic polynomial of P_{a}D_{χ} is r(t) = (t^{c} − ϵ)^{\frac{n}{c}}.

(c) Assume char(k) |̸ (\frac{n}{c})!. Then σ^{(i)}(P_{a}D_{χ}) ≠ 0 for any i divisible by c.

Proof. (a) The identity (P_{a}D_{χ})^{c} = ϵI_{n}, where ϵ = χ(a)^{\frac{1}{2} c(c−1)}, is immediate from (3.2). To see that ϵ = 1 or −1, note that ϵ^{2} = (χ(a)^{c})^{c−1} = 1^{c−1} = 1.

(b) Let C be the cyclic subgroup of A × A^{∗} generated by (a, χ), so that c = |C|. For each α ∈ (A × A^{∗})/C, let V_{α} be the vector subspace of M_{n} spanned by (b, η) ∈ α. Each V_{α} is a c-dimensional subspace of M_{n}, which is stable under right multiplication by P_{a}D_{χ}. Since the matrices P_{a}D_{χ} form a basis of M_{n} as (a, χ) ranges over A × A^{∗} (see Lemma 3.2), we can write

M_{n} = ⨁_{α∈(A×A^{∗})/C} V_{α} .  (3.4)

By part (a), (P_{a}D_{χ})^{c} = ϵI_{n}. It is now easy to see that the characteristic polynomial for the action of P_{a}D_{χ} on each V_{α} is p(t) = t^{c} − ϵ. Consequenly, the charactersistic polynomial for the left multiplication action of P_{a}D_{χ} on M_{n} is q(t) = p(t)^{n^{2}/c} (one factor of p(t) for each subspace V_{α} in (3.4)), and the characteristic polynomial of the n × n-matrix P_{a}D_{χ} (or, equivalently, of its action on n × 1-column vectors) is

r(t) = q(t)^{1/n} = p(t)^{n/c} = (t^{c} − ϵ)^{n/c} ,

as claimed.

(c) The binomial formula tells us that under our assumption on char(k), every monomial of the form t^{n−i} with i divisible by c (and i ≤ n), appears in r(t) with a nonzero coefficient. In other words, for these values of i, σ^{(i)}(P_{a}D_{χ}) ≠ 0, as claimed.

4. Proof of Theorem 1.5

We may (and will, throughout this section) assume without loss of generality that k is an algebraically closed field. Otherwise we can simply replace D by \overline{D} = D⊗_{k} \overline{k}, where \overline{k} is the algebraic closure of k: if the system (1.5) has no nontrivial solutions in \overline{D}, it cannot have one in D.

Our goal is to deduce Theorem 1.5 as a special case of Proposition 2.4. We shall now proceed to introduce the finite abelian group H and the PGL_{n}-varieties X and Y and to show that they satisfy the conditions of Proposition 2.4. We will then apply Proposition 2.4 with these H, X, and Y, to conclude that the system (1.5) has no nontrivial solutions in D_{n} or UD(n).

The group H. We define H to be the finite abelian subgroup of PGL_{n} given by

H = A × A^{∗} ↪^{ϕ_{A}} PGL_{n} , where A = ℤ/p_{1}ℤ × ⋯ × ℤ/p_{s}ℤ .  (4.1)

Here, as in Section 1, n = p_{1} … p_{s}, where p_{1}, …, p_{s} are not necessarily distinct primes; the inclusion ϕ_{A} is as in (3.3). Note that the assumption char(k) |̸ n! of Theorem 1.5 implies that |H| = n^{2} is prime to char(k).

The variety X. We shall now write the algebras that come up in the statement of Theorem 1.5, namely D = UD(n) and D = D_{n}, in the form RMaps_{PGL_{n}}(X, M_{n}) for specific PGL_{n}-varieties X. Note that we do not assume char(k) = 0.

Lemma 4.1. (Procesi) UD(n) = RMaps_{PGL_{n}}(X, M_{n}), where X = (M_{n})^{2} and PGL_{n} acts on X by simultaneous conjugation.

Proof. See [Sa, Theorem 14.16], cf. also [P2, Theorem 2.1] or [RY_{2}, Example 3.1].

Let G be an algebraic group, S be a closed subgroup of G, and Y be an affine S-variety. The groups S and G act on G × Y via respectively, s(g, y) = (gs^{−1}, sy) and g′(g, y) = (g′g, y); moreover, the two actions commute. Thus the quotient (G × Y)//S = Spec(k[G × Y]^{S}) is a G-variety; we will denote it by G ∗_{S} Y. We will restrict our attention to the case where S is a finite group of order prime to char(k). In this case a theorem of Hilbert and Noether (see, e.g., [Sm, Theorem 1.1]) tells us that k[G × Y]^{S} is a finitely generated k-algebra, i.e., G ∗_{S} Y is again an affine variety (of finite type).

Lemma 4.2. There exists a faithful 2s-dimensional linear representation V of H such that D_{n} ≃ RMaps_{PGL_{n}}(X, M_{n}), where X = PGL_{n} ∗_{H} V.

Proof. Choose a set of generators a_{1}, …, a_{s} for A and a “dual” set of generators χ_{1}, …, χ_{s} for A^{∗} so that

χ_{i}(a_{j}) = \{ \begin{matrix}1 & if i ≠ j \\ ζ_{p_{i}} & if i = j , \end{matrix}

where ζ_{p_{i}} is the same primitive p_{i}th root of unity used in defining (z_{i}, w_{i})_{p_{i}}; see (1.3) and (1.4). Consider the faithful action of H = A × A^{∗} on V = k^{2s} given by

(a, χ) : (α_{1}, …, α_{s}, β_{1}, …, β_{s}) ↦
(χ^{−1}(a_{1})α_{1}, …, χ^{−1}(a_{s})α_{s}, χ_{1}(a)β_{1}, …, χ_{s}(a)β_{s}) .

Set X = PGL_{n} ∗_{H} V and R = RMaps_{PGL_{n}}(X, M_{n}). Note that

k(X)^{PGL^{n}} = k(PGL_{n} × V)^{PGL^{n}×H} = k(V)^{H} = k(α_{1}^{p_{1}}, β_{l}^{p_{1}}, …, α_{s}^{p_{s}}, β_{s}^{p_{s}}) .(4.2)

Define elements π_{i} and η_{i} of R by

\begin{matrix}π_{i}: [g, (α_{1}, …, α_{s}, β_{1}, …, β_{s})] & ↦ α_{i}gP_{a_{i}}g^{−1} \\ η_{i}: [g, (α_{1}, …, α_{s}, β_{1}, …, β_{s})] & ↦ β_{i}gD_{χ_{i}}g^{−1} . \end{matrix}  (4.3)

These elements are well-defined because π_{i}(g, v) = π_{i}(gh^{−1}, hv) and η_{i}(g, v) = η_{i}(gh^{−1}, hv) for every h ∈ H and i = 1, …, s; see (3.2). Note that since P_{a_{i}} and D_{χ_{i}} generate M_{n}(k) as a k-algebra, as i ranges from 1 to n (cf. Lemma 3.2), there exists a dense Zariski dense open subset X_{0} ⊂ X such that

π_{i}(x) and η_{i}(x) generate M_{n}(k) for every x ∈ X_{0}.  (4.4)

In particular, if f is a central element of R then f(x) is a scalar matrix for every x ∈ X_{0}. Consequently, the center Z(R) consists of rational maps X −→ M_{n} whose image lies in the subspace of scalar matrices. In other words,

Z(R) = k(X)^{PGL^{n}}  (4.5)

where, as before, we identify f ∈ k(X)^{PGL^{n}} with the PGL_{n}-equivariant rational map X −→ M_{n}(k) given by x ↦ f(x)I_{n}.

We are now ready to construct an isomorphism between D_{n} and R. First we identify D_{n} with the skew-polynomial ring

D_{n} = Z(R){x_{1}, y_{1}, …, x_{s}, y_{s}} ,

where x_{i}^{p_{i}} = α_{i}^{p_{i}}, y_{i}^{p_{i}} = β_{i}^{p_{i}}, y_{i}x_{i} = ζ_{p_{i}}x_{i}y_{i} and all other pair of variables commute. (Recall that Z(R) is the purely transcendental extension of k generated by α_{1}^{p_{1}}, β_{1}^{p_{1}}, …, α_{s}^{p_{s}}, β_{s}^{p_{s}}; see (4.2) and (4.5).) Let ϕ: D_{n} ⟶ R be the Z(R)-algebra homomorphism given by ϕ(x_{i}) = π_{i} and ϕ(y_{i}) = η_{i}. This homomorphism is welldefined because π_{i} and η_{i} satisfy the same relations as x_{i} and y_{i}; see (4.3) and (3.2).

We claim ϕ is an isomorphism. Indeed, ϕ is injective since D_{n} is a simple algebra. Moreover, since dim_{k}(M_{n}) = n^{2}, it is easy to see that dim_{Z(R)} R ≤ n^{2} (see, e.g. , [Re_{2}, Lemma 7.4(a)] for a characteristic-free proof). This shows that ϕ is an isomorphism and thus completes the proof of Lemma 4.2.

The variety Y. We now define the PGL_{n}-variety Y by

Y = \{(y_{1} : ⋯: y_{m}) ∈ ℙ((M_{n})^{m}) | \begin{matrix}σ^{(i)}(y_{1}) = ⋯ = σ^{(i)}(y_{m}) \\ σ^{(j)}(y_{1} … y_{m}) = 0 \end{matrix} \} ,  (4.6)

as in Proposition 2.4. Recall that our goal is to use Proposition 2.4 to show that the system (1.5) has no nontrivial solutions.

Lemma 4.3. Under the assumptions of Theorem 1.5 (i.e., char(k) |̸ n!, sqf(n) | m and sqf(n) | i), H acts on Y without fixed points.

Proof. The H-fixed points in ℙ((M_{n})^{m}) are of the form y = (y_{1} : ⋯: y_{m}), where each y_{i} is either 0 or an element of M_{n} which spans a 1-dimensional character space for H. Moreover, the associated characters of all non-zero y_{i} have to be the same. Thus, in view of Lemma 3.2, there exists an element (a, χ) ∈ A × A^{∗} such that y_{i} = t_{i}P_{a}D_{χ} for some t_{1}, …, t_{m} ∈ k. Note that at least one t_{i} has to be non-zero, since otherwise y = (0 : ⋯: 0) is not a well-defined point of ℙ((M_{n})^{m}).

Now suppose y is an H-fixed point of Y. Substituting y_{i} = t_{i}P_{a}D_{χ} into the defining equations for Y, we obtain

\{ \begin{matrix}t_{1}^{i}σ^{(i)}(P_{a}D_{χ}) = ⋯ = t_{m}^{i}σ^{(i)}(P_{a}D_{χ}) , \\ t_{1} … t_{m}σ^{(j)}((P_{a}D_{χ})^{m}) = 0 . \end{matrix}  (4.7)

Let c be the order of (a, χ) in A×A^{∗}. Then c | exp(A), exp(A) = sqf(n), sqf(n) | m, sqf(n) | i, and thus, c | m and c | i. By Lemma 3.3(a), (P_{a}D_{χ})^{m} = ±I_{n}, and hence, σ^{(j)}((P_{a}D_{χ})^{m}) ≠ 0. By Lemma 3.3(c), σ^{(i)}(P_{a}D_{χ}) ≠ 0. Therefore, we can rewrite (4.7) as

\{ \begin{matrix}t_{1}^{i} = ⋯ = t_{m}^{i} , \\ t_{1} … t_{m} = 0 . \end{matrix}

This system has no solutions other than t_{1} = ⋯ = t_{m} = 0, a contradiction. We conclude that Y has no H-fixed points, as claimed.

Conclusion of the proof. In order to complete the proof, it remains to show that X has a smooth H-fixed point; the desired conclusion will then follow by applying Proposition 2.4 to the abelian group H and PGL_{n}-varieties X and Y we introduced above.

If D = UD(n) then X = (M_{n})^{2} (see Lemma 4.1), and the origin is a smooth H-fixed point of X.

If D = D_{n} then X = PGL_{n} ∗_{H} V = (PGL_{n} × V)//H; see Lemma 4.2. Since PGL_{n} ×V is a smooth variely, and H acts freely on it, X is also smooth. Moreover, the point of X represented by (1, 0) ∈ PGL_{n} × V, is clearly fixed by H. Thus X has a smooth H-fixed point, as claimed.

This completes the proof of Theorem 1.5.

Refinements. A slight modification of the above argument proves the following more general variant of Theorem 1.5.

Theorem 4.4. Let P (z_{1}, …, z_{v}) ∈ k{z_{1}, …, z_{v}} be a homogeneous (noncommutative) polynomial of degree d in v variables. The system of equations

\{ \begin{matrix}σ^{(i)}(x_{1}^{u}) = ⋯ = σ^{(i)}(x_{v}^{u}) \\ σ^{(j)}(P (x_{1}, …, x_{v})) = 0 \end{matrix}  (4.8)

has no nontrivial solutions in D_{n} or UD(n), provided that (i) iu and jd are divisible by sqf(n). (ii) P (ζ_{1}, …, ζ_{v}) ≠ 0 for any (not necessarily primitive) ij-th roots of unity

ζ_{1}, …, ζ_{v}.

Note that if we set u = 1, d = v = m and P (z_{1}, …, z_{v}) = z_{1} … z_{v}, then we recover Theorem 1.5 from Theorem 4.4.

Remark 4.5. Suppose K = k(a_{1}, b_{2}, …, a_{l}, b_{l}) and

D = (a_{1}, b_{1})_{r_{1}} ⊗_{K} ⋯ ⊗_{K} (a_{l}, b_{l})_{r_{l}}

be a tensor product of generic symbol algebras of degree n = r_{1} … r_{l}. Denote the least common multiple of r_{1}, …, r_{l} by e. (Equivalently, e is the exponent of D.) Then the system (1.5) has no solutions in D as long as i and m are divisible by e. The proof is the same as above, except that instead of choosing H and A as in (4.1), we take H = ϕ_{A}(A × A^{∗}) with A = (ℤ/r_{1}ℤ) × ⋯ × (ℤ/r_{l}ℤ). Similarly, the system (4.8) has no solutions in D, provided that iu and jd is divisible by e, and condition (ii) of Theorem 4.4 holds.

Remark 4.6. Theorem 1.5 remains true if D is replaced by the general field extension L_{n}/K_{n}. The reason is that there is a natural embedding α: L_{n} ↪ UD(n) such that

α: σ_{L_{n}/K_{n}}^{(i)}(y) ↦ σ_{UD(n)/Z(n)}^{(i)}(α(y))

for every y ∈ L_{n} and every i = 1, …, n. Indeed, recall that UD(n) is generated by two generic n × n-matrices, X = (s_{ij}) and Y = (t_{ij}): we can define α(x) = X and α(a_{i}) = σ^{(i)}(X), see, e.g., [P1, Lemma II.1.4]. If system (1.5) had a nontrivial solution in L_{n}, it would then have a nontrivial solution in UD(n), contradicting Theorem 1.5.

Remark 4.7. Suppose char(k) = 0, n = p^{r} and D′ as a prime-to-p extension of D_{n} or UD(n). Then Theorem 1.5, Corollary 1.6 and Theorem 4.4 remain valid if D is replaced by D′. Indeed, let X be as in Lemma 4.1 (if D = UD(n)) and Lemma 4.2 (if D = D_{n}). Then we can write D′ as RMaps_{PGL_{n}}(X′, M_{n}), where X′ −→ X is a PGL_{n}-invariant rational cover, of degree prime to p. We may assume that X′ is smooth and projective. (This follows from canonical resolution of singularities; see [RY_{2}, Proposition 2.2].) Since H is a p-group, the Going Up Theorem says that X′ has an H-fixed point; see [RY_{1}, Proposition A.4]. The desired conclusion now follows from Proposition 2.4.

5. Proof of Theorem 1.7

We may assume without loss of generality that k is an algebraically closed field; otherwise we may simply replace K_{n} and L_{n} by K_{n} ⊗_{k} \overline{k} and L_{n} ⊗_{k} \overline{k} respectively, where \overline{k} is the algebraic closure of k.

Let f(x) = x^{n} + a_{1}x^{n−1} + ⋯ + a_{n} and L_{n} = K_{n}[x]/(f(x)), as in (1.6). The normal closure of L_{n} over K_{n} is the field L′ = K_{n}(x_{1}, …, x_{n}) = k(x_{1}, …, x_{n}), where x_{1}, …, x_{n} are the roots of f; they are algebraically independent over k. We will identify L_{n} with K_{n}(x_{1}) by identifying x ∈ L_{n} with x_{1} ∈ k(x_{1}, …, x_{n}) and a_{i} with (−1)^{i}s_{i}(x_{1}, …, x_{n}), where s_{i} is the ith elementary symmetric polynomial.

We shall deduce Theorem 1.7 as a particular case of Proposition 2.2, with m = 1, K = K_{n}, L = L_{n}, L′ as above, and G = Gal(L′/L_{n}) = S_{n}. We will now define the remaining objects that appear in the statement of Proposition 2.2, namely the abelian subgroup H of G = S_{n} and the G-varieties X and Y.

We set H = H_{1} × H_{2}, with H_{1} = ψ_{A_{1}}(A_{1}) ⊂ S_{n_{1}}, H_{2} = ψ_{A_{2}}(A_{2}) ⊂ S_{n_{2}}, as in (3.1); here for i = 1, 2, A_{i} is an abelian subgroup of order n_{i} and exponent sqf(n_{i}). More precisely, if n_{1} = p_{1} … p_{s} and n_{2} = q_{1} … q_{t} are written as products of (not necessarily distinct) primes then

H_{1} ≃ A_{1} = (ℤ/p_{1}ℤ) × ⋯ × (ℤ/p_{s}ℤ)
and  (5.1)
H_{2} ≃ A_{2} = (ℤ/q_{1}ℤ) × ⋯ × (ℤ/q_{t}ℤ) .

We define X = 𝔸^{n}, with the natural permutation action of G = S_{n}. If we denote the coordinates on 𝔸^{n} by x_{1}, …, x_{n} then k(X) = k(x_{1}, …, x_{n}) = L′ as fields with S_{n}-action. The origin is a smooth point of X fixed by S_{n} and, hence, by H.

The S_{n}-variety Y is defined as the subvariety of ℙ(𝔸^{n}) = ℙ^{n−1} given by

\{ \begin{matrix}x_{1}^{m_{1}} + ⋯ + x_{n}^{m_{1}} & = 0 \\ x_{1}^{m_{2}} + ⋯ + x_{n}^{m_{2}} & = 0 . \end{matrix}  (5.2)

In order to apply Proposition 2.2, it is now sufficient to prove the following:

Lemma 5.1. Under the assumptions (i) and (ii) of Theorem 1.7, Y has no H-fixed points.

Proof. By Lemma 3.1, the fixed points y for the H-action on ℙ^{n−1} = ℙ^{n_{1}+n_{2}−1} are of one of the following three types:

Type I: y = R_{a,b} = (\underbrace{a : ⋯: a}_{n_{1} times} : \underbrace{b : ⋯: b}_{n_{2} times}), for some a, b ∈ k, not both 0.

Type II: y = (R_{χ}, 0) = (χ(α_{1}) : ⋯: χ(α_{n_{1}}) : 0 : ⋯: 0), where H_{1} = {α_{1}, …, α_{n_{1}}} and χ is a character of H_{1}.

Type III: y = (0, R_{η}) = (0 : ⋯: 0 : η(β_{1}) : ⋯: η(β_{n_{2}})), where H_{2} = {β_{1}, …, β_{n_{2}}} and η is a character of H_{2}.

Consider a point of type I. Substituting the coordinates of R_{a,b} into (5.2), we see that R_{a,b} lies in Y if and only if (a, b) is a nontrivial solution of the homogeneous system

\{ \begin{matrix}n_{1}a^{m_{1}} + n_{2}b^{m_{1}} = 0 \\ n_{1}a^{m_{2}} + n_{2}b^{m_{2}} = 0 . \end{matrix}  (5.3)

An elementary computation shows that under assumption (i) of Theorem 1.7 this system has no nontrivial solutions. Hence we conclude that no point of type I can lie on Y.

We now turn to points of types II and III. Since H_{1} has exponent sqf(n_{1}), we see that χ(α_{i})^{sqf(n_{1})} = 1 for every α_{i} ∈ H_{1}. It follows from the assumptions of Theorem 1.7 that n_{1} ≠ 0 in k and either m_{1} or m_{2} is divisible by sqf(n_{1}); consequently, (R_{χ}, 0) does not lie on Y. Similarly, (0, R_{η}) does not lie on Y. Hence, no point of type II or III lies on Y. This completes the proof of the lemma and thus of Theorem 1.7.

Remark 5.2. Theorem 1.7 fails if the field extension L_{n}/K_{n} is replaced by the generic division algebra UD(n). Suppose, for simplicity, that k is an algebraically closed field of characteristic zero. Then, by a theorem of Wedderburn, UD(3) is cyclic; thus it has an elements x and y such that x = ζ_{3}yxy^{−1}, where ζ_{3} is a primitive cube root of 1. It is now easy to see that tr(x) = tr(x^{2}) = 0. On the other hand, Theorem 1.7 with n_{1} = m_{1} = 1 and n_{2} = m_{2} = 2, says that no such element can exist in L_{3}.

Another example of this kind can be constructed for n = 6. The algebra D = UD(6) is known to be cyclic; hence, it has a non-zero element z such that tr(z^{i}) = 0 for i = 1, …, 5. On the other hand, Theorem 1.7 says that the systems tr(x) = tr(x^{5}) = 0 or tr(x^{2}) = tr(x^{4}) = 0 have no solutions in L_{6}^{∗}.

Remark 5.3. Let n_{1} = p_{1} … p_{s} and n_{2} = q_{1} … q_{t}, where p_{1}, …, p_{s}, q_{1}, …, q_{t} are (not necessarily distinct) primes. Suppose z_{1}, …, z_{s} and w_{1}, …, w_{t} are independent variables over k. Set E_{1} = k(z_{1}, …, z_{s}, w_{1}^{q_{1}}, …, w_{t}^{q_{t}}), E_{2} = k(z_{1}^{p_{1}}, …, z_{s}^{p_{s}}, w_{1}, …, w_{t}), and F = k(z_{1}^{p_{1}}, …, z_{s}^{p_{s}}, w_{1}^{q_{1}}, …, w_{t}^{q_{t}}). Then we can replace L_{n}/K_{n} by the n-dimensional etale F -algebra E = E_{1}⊕E_{2} (cf. [Re_{1}, Section 4]) in the statement of Theorem 1.7. In other words,

under assumptions (i) and (ii) of Theorem 1.7 the system of equations tr(x^{m_{1}}) = tr(x^{m_{2}}) = 0 has no nontrivial solutions in E. The role played by E in this setting is analogous to the role played by D_{n} in the

setting of Theorem 1.5. In particular, one can show that E = RMaps_{S_{n}}(X, 𝔸^{n}), where X = S_{n} ∗_{H} V, V is a faithful (s + t)-dimensional linear representation of H = H_{1} × H_{2}, and the algebra structure on RMaps_{S_{n}}(X, 𝔸^{n}) is induced from the algebra structure on 𝔸^{n} = \underbrace{k ⊕ ⋯ ⊕ k}_{n times} (compare with Lemma 4.2). Since X has a

smooth H-fixed point (namely, the point represented by (id, 0) ∈ S_{n} × V), the rest of our argument goes through unchanged.

6. Systems of the form σ^{(m_{1})}(x) = σ^{(m_{2})}(x) = 0

We do not know whether or not the system tr(x^{m_{1}}) = tr(x^{m_{2}}) = 0 may be replaced by the system

σ^{(m_{1})}(x) = σ^{(m_{2})}(x) = 0 .  (6.1)

in the statement of Theorem 1.7. (Such a result would be of interest, since it would mean that the general polynomial of degree n cannot be transformed, by a Tschirnhaus substitution, into a polynomial t^{n} + b_{1}t^{n−1} + ⋯ + b_{n}, with b_{m_{1}} = b_{m_{2}} = 0.) Every step of our proof of Theorem 1.7 goes through in this case, except that the system (5.3) is replaced by the system

\{ \begin{matrix}s_{m_{1}}(a, …, a, b, …, b) = 0 \\ s_{m_{2}}(a, …, a, b, …, b) = 0 , \end{matrix}  (6.2)

where (a, …, a, b, …, b) stands for (\underbrace{a, …, a}_{n_{1} times}, \underbrace{b, …, b}_{n_{2} times}) and s_{i} denotes the ith ele-

mentary symmetric polynomial. Thus:

Proposition 6.1. Let n_{1} and n_{2} be positive integers prime to char(k), and L_{n}/K_{n} be the general field extension of degree n = n_{1} + n_{2}. Then the system (6.1) has no nontrivial solutions x ∈ L_{n}^{∗}, provided that each sqf(n_{i}) (i = 1, 2) divides m_{1} or m_{2} and the system (6.2) has no nontrivial solutions (a, b) ∈ k^{2}.

Of course, this result is less satisfying than Theorem 1.7 because we do not know for what values of n_{1}, m_{1}, n_{2} and m_{2} the system (6.2) has no nontrivial solutions. (The analogous question for the system (5.3) is quite easy: the answer is given by condition (i) of Theorem 1.7.) Nevertheless, for low values of n, Proposition 6.1 gives us a rather complete picture. We shall give two such examples below.

Before preceeding with the examples, we record a simple observation.

Remark 6.2. Let E/F be a field extension of degree n. Multiplying (1.1) by det((λx)^{−1}), we easily obtain the identity σ^{(n−i)}(x^{−1}) = σ^{(i)}(x)/σ^{(n)}(x). In particular, if x ∈ E satisfies (6.1) then σ^{(n−m_{1})}(x^{−1}) = σ^{(n−m_{2})}(x^{−1}) = 0.

Example 6.3. Let L_{5}/K_{5} be the general field extension of degree 5 and let 1 ≤ m_{1} < m_{2} ≤ 5. Then the system (6.1) has a nontrivial solution x ∈ L_{5}^{∗} if and only if (m_{1}, m_{2}) = (1, 3) or (2, 4).

Proof. By the theorem of Hermite cited in Example 1.1, the system (6.1) has a solution 0 ≠ x ∈ L_{5} for (m_{1}, m_{2}) = (1, 3). Then x^{−1} is a solution to (6.1) with (m_{1}, m_{2}) = (2, 4); see Remark 6.2.

It remains to show that there are no solutions for any other values of m_{1} and m_{2}. Indeed, we may assume without loss of generality that m_{2} ≠ 5, since σ^{(5)}(x) = − det(x) ≠ 0 for any x ∈ L_{5}^{∗}. The remaining possibilities for (m_{1}, m_{2}) are: (1, 2), (1, 4), (2, 3), and (3, 4). In view of Remark 6.2, we only need to consider (1, 2), (1, 4) and (2, 3).

(m_{1}, m_{2}) = (1, 2). By Newton’s formulas the system σ^{(1)}(x) = σ^{(2)}(x) = 0 is equivalent to tr(x) = tr(x^{2}) = 0. The latter system has no solutions by Theorem 1.7 with n_{1} = 1 and n_{2} = 4. (Alternatively, use Proposition 6.1 with n_{1} = 1, n_{2} = 4 or appeal to [Re_{1}, Theorem 1.3(b)], with p = 2 and m = 2.)

(m_{1}, m_{2}) = (1, 4). Apply Proposition 6.1 with n_{1} = 1 and n_{2} = 4. In this case (6.2) reduces to

\{ \begin{matrix}s_{1}(a, b, b, b, b) = a + 4b = 0 \\ s_{4}(a, b, b, b, b) = b^{4} + 4ab^{3} = 0 . \end{matrix}

It is easy to see that this system has no nontrivial solutions. (Alternatively, use [Re_{1}, Theorem 6.1b].)

(m_{1}, m_{2}) = (2, 3). Apply Proposition 6.1 with n_{1} = 2 and n_{2} = 3. In this case (6.2) becomes

\{ \begin{matrix}s_{2}(a, a, b, b, b) = a^{2} + 6ab + 3b^{2} = 0 \\ s_{3}(a, a, b, b, b) = 3a^{2}b + 6ab^{2} + b^{3} = 0 . \end{matrix}

This system has no nontrivial solutions.

Example 6.4. Let L_{6}/K_{6} be the general field extension of degree 6 and let 1 ≤ m_{1} < m_{2} ≤ 6. Then the system (6.1) has a nontrivial solution x ∈ L_{5}^{∗} if and only if (m_{1}, m_{2}) = (1, 3) or (3, 5).

Proof. The existence of solutions for (m_{1}, m_{2}) = (1, 3) and (3, 5) follows from Example 1.1 and Remark 6.2.

We may assume m_{2} ≤ 5 because σ^{(6)}(x) = det(x) ≠ 0 for any x ∈ L_{6}^{∗}. It is now enough to show that there are no solutions for (m_{1}, m_{2}) = (1, 2), (1, 4), (1, 5), (2, 3), and (2, 4); the remaining cases follow from these by Remark 6.2.

(m_{1}, m_{2}) = (1, 2). In this case (6.2) is equivalent to tr(x) = tr(x^{2}) = 0. The latter system has no solutions by Theorem 1.7 with n_{1} = 2 and n_{2} = 4. (Alternatively, use Proposition 6.1 with n_{1} = 1, n_{2} = 4 or appeal to [Re_{1}, Theorem 1.3(c)], with p = 2, m = 2 and l = 1.)

(m_{1}, m_{2}) = (1, 4). Apply Proposition 6.1 with n_{1} = 2, n_{2} = 4. In this case (6.2) reduces to 2a + 4b = 6a^{2}b^{2} + 8ab^{3} + b^{4} = 0. This system has no nontrivial solutions.

(m_{1}, m_{2}) = (1, 5). Apply Proposition 6.1 with n_{1} = 1, n_{2} = 5. In this case (6.2) reduces to a+5b = 5ab^{4} +b^{5} = 0. There are no nontrivial solutions. (Alternatively, use [Re_{1}, Theorem 1.3(b)] with p = 5.)

(m_{1}, m_{2}) = (2, 3). Apply Proposition 6.1 with n_{1} = 2, n_{2} = 4. In this case (6.2) becomes a^{2} + 8ab + 6b^{2} = 4a^{2}b + 12ab^{2} + 4b^{3} = 0. There are no nontrivial solutions.

(m_{1}, m_{2}) = (2, 4). Use Proposition 6.1 with n_{1} = 2, n_{2} = 4. In this case (6.2) becomes a^{2} +8ab+6b^{2} = 6a^{2}b^{2} +8ab^{3} +b^{4} = 0. Once again, there are no nontrivial solutions.

7. A further generalization

In this section we will show that the assumption that the G-variety Y in Proposition 2.2 has no fixed points can sometimes be weakened. We will present a general result extending Proposition 2.2 and illustrate it with an example. One can generalize Proposition 2.4 in a similar manner; we leave the details to an interested reader.

In this section we assume that k is algebraically closed.

Proposition 7.1. Assume

(i) L/K is a separable field extension of degree n, L′ is the normal closure of L over K, G = Gal(L′, K), and H is an abelian subgroup of G,

(ii) Y ⊃ Z are subvarieties of (𝔸^{n})^{m} given, respectively, by systems of Ginvariant polynomial equations P_{1} = ⋯ = P_{s} = 0 and Q_{1} = ⋯ = Q_{r} = 0,

(iii) there exists a complete H-variety W without H-fixed points and a regular H-equivariant map h: Y − Z ⟶ W, and

(iv) there exists a G-variety X such that such that k(X) = L′ as fields with G-action, and X has a smooth H-fixed point.

Then any solution (a_{1}, …, a_{m}) ∈ L^{m} of the system

P_{1}(x_{1}, …, x_{m}) = ⋯ = P_{s}(x_{1}, …, x_{m}) = 0  (7.1)

also satisfies the system

Q_{1}(x_{1}, …, x_{m}) = ⋯ = Q_{r}(x_{1}, …, x_{m}) = 0 .  (7.2)

Note that since Z ⊂ Y, the ideal (Q_{1}, …, Q_{r}) ⊂ k[(𝔸^{n})^{m}] contains some power of the ideal (P_{1}, …, P_{s}). Hence, any solution of (7.2) in L^{n} is a solution of (7.1). Proposition 7.1 asserts that under assumptions (i)–(iv), the opposite is also true.

Proof. Given a solution (a_{1}, …, a_{m}) of (7.1), we construct a rational map f: X −→ Y ⊂ (𝔸^{n})^{m}, as in the proof of Proposition 2.2. If (a_{1}, …, a_{m}) does not satisfy (7.2), then f(X) ⊄ Z and hence, the composition X −→^{f} Y −→^{h} W is a well-defined H-equivariant rational map. As X has a smooth H-fixed point, Theorem 2.1 says that W also has one, a contradiction.

Remark 7.2. To see that Proposition 2.2 is a special case of Proposition 7.1, assume that the polynomials P_{1}, …, P_{s} are homogeneous, so that Y is a cone in (𝔸^{n})^{m}, and Z is the origin in (𝔸^{n})^{m}. Note that the origin of (𝔸^{n})^{m} can be cut out by G-invariant homogeneous polynomials (this is true for any finite group representation), thus we can choose Q_{1}, …, Q_{r} ∈ k[(𝔸^{n})^{m}]^{G} to be generators of the ideal of the origin in k[(𝔸^{n})^{m}].

Let W ⊂ ℙ((𝔸^{n})^{m}) be the projectivisation of the cone Y, and h: Y − Z ⟶ W the natural projection. If W has no H-fixed points, and X has a smooth H-fixed point then Proposition 7.1 implies that the system (7.1) has no solutions, except for x_{1} = ⋯ = x_{m} = 0. This is precisely the statement of Proposition 2.2.

Remark 7.3. Proposition 7.1 can be applied in the following situation. Suppose that Z is the singular set of Y. Let Ỹ be the closure of Y ⊂ (𝔸^{n})^{m} = 𝔸^{nm} in ℙ^{nm} ⊃ 𝔸^{nm}; note that the G-action on (𝔸^{n})^{m} extends to a regular G-action on ℙ^{nm}, and Ỹ is G-invariant. Let π: W ⟶ Ỹ be the canonical resolution of singularities. Such a resolution is known to exist if char(k) = 0; see the discussion and the references in [RY_{1}, Section 3]. Note that π is an isomorphism over Y − Z and thus we can take h = π^{−1} : Y − Z ⟶ W. If W has no H-fixed points then Proposition 7.1 applies.

Example 7.4. Suppose n is prime and n ≠ char(k). Then for any c ∈ k the equation

∑_{i=1}^{n−1} σ^{(i)}(x)^{n}σ^{(n)}(x)^{n−1−i} + cσ^{(n)}(x)^{2n−2} = 0 ,  (7.3)

has no nontrivial solutions in the general field extension L_{n}/K_{n}; see (1.6). Here σ^{(i)} stands for σ_{L_{n}/K_{n}}^{(i)}.

Proof. We may assume without loss of generality that k is algebraically closed, and thus, contains the roots of unity.

First consider the case c ≠ 0. We apply Proposition 7.1 in the following setting: K = K_{n}, L = L_{n}, G = S_{n}, X = 𝔸^{n} with the natural S_{n}-action, H = the cyclic subgroup of S_{n} generated by the n-cycle h = (1 2 … n), s = m = 1, and P_{1} = s_{1}^{n}s_{n}^{n−2}+s_{2}^{n}s_{n}^{n−3}+⋯+s_{n−1}^{n}+cs_{n}^{2n−2}, where s_{i} denotes the ith elementary symmetric polynomial in the coordinates x_{1}, …, x_{n} in 𝔸^{n}. (To construct P_{1}, we replaced σ^{(i)}(x) by (−1)^{i}s_{i}(x_{1}, …, x_{n}) in the left hand side of 7.3.) Note that P_{1} is not homogeneous in x_{1}, …, x_{n} as c ≠ 0.

We take Z to be the origin in 𝔸^{n}. Similarly to Remark 7.3, let Ỹ the closure of Y ⊂ A^{n} in ℙ^{n}; then the H-action on Ỹ − Z is free. Let ℙ̃^{n} ⟶ ℙ^{n} be the blowup of Z; we identify its exceptional divisor S with ℙ^{n−1}. Let Y′ be the strict transform of Ỹ; then Y′ ⟶ Ỹ is a blowup centered at Z, and S ∩Y′ is the hypersurface in ℙ^{n−1} given by the homogeneous equation \overline{P}_{1} = 0 where \overline{P}_{1} = s_{1}^{n}s_{n}^{n−2}+s_{2}^{n}s_{n}^{n−3}+⋯+s_{n−1}^{n} is the initial form of P_{1}.

The intersection S ∩ Y′ contains H-fixed points q_{ζ} = (1 : ζ : ζ^{2} : ⋯: ζ^{n−1}) for each n-th root of unity ζ ≠ 1. Let W ⟶ Y′ be the blowup of these n − 1 points. We claim that W has no H-fixed points.

To see this, consider the hypersurfaces S_{i} ⊂ ℙ̃^{n} for i = 1, …, n − 1 which are the closures in ℙ̃^{n} of the hypersurfaces in 𝔸^{n}−Z given by the equations s_{i} = 0. For each i, the intersection S_{i} ∩S is the hypersurface in S = ℙ^{n−1} given by the homogeneous equation s_{i} = 0; in particular, each S_{i} passes through q_{ζ}. Consider the (n − 1) × (n − 1) Jacobian determinants D_{l}(q_{ζ}) = det(∂s_{i}/∂x_{j})(q_{ζ}), where i = 1, …, n − 1 and j = 1, …, l̂, …, n. By Newton’s formulas D_{l}(q_{ζ}) = det(∂p_{i}/∂x_{j})(q_{ζ}), where p_{i} = x_{1}^{i} + ⋯ + x_{n}^{i}. The latter determinant is a Vandermonde determinant, which does not vanish at q_{ζ}. This shows that the hypersurfaces S_{i} ∩ S are smooth and intersect transversely (in S = ℙ^{n−1}) at each q_{ζ}; hence S_{1}, …, S_{n−1} and S are smooth and intersect transversely (in ℙ̃^{n}) at each q_{ζ}.

Thus the tangent spaces T_{q_{ζ}}(S_{1}), …, T_{q_{ζ}}(S_{n−1}), together with T_{q_{ζ}}(S), form a system of coordinate hyperplanes in T_{q_{ζ}}(ℙ̃^{n}). Since each S_{i} is H-invariant, the linear H-action on T_{q_{ζ}}(ℙ̃^{n}) is diagonalized in this coordinate system. The group H acts by different characters on each of the coordinate directions; in fact, h acts by multiplication by ζ^{i} on T_{q_{ζ}}(ℙ̃^{n})/T_{q_{ζ}}(S_{i}), and trivially on T_{q_{ζ}}(ℙ̃^{n})/T_{q_{ζ}}(S). Identifying the exceptional divisor E_{q_{ζ}} of the blowup of ℙ̃^{n} centered at q_{ζ}, with ℙ(T_{q_{ζ}}(ℙ̃^{n})), we see that the H-fixed points on E_{q_{ζ}} are the points of ℙ(T_{q_{ζ}}(ℙ̃^{n})) that correspond to the directions of the coordinate axes in T_{q_{ζ}}(ℙ̃^{n}). The exceptional divisor of W over q_{ζ} is the projectivisation of the tangent cone to Y′ at q_{ζ}, and the latter does not contain the coordinate axes. We conclude that W does not have H-fixed points, as claimed.

Thus, we may apply Proposition 7.1; it shows that the equation (7.3) has no nontrivial solutions, similarly to Remark 7.2.

In case c = 0, we need to make the following changes. Now Y is an affine cone; we take Z to be the union of (n − 1)! lines that correspond to the points (ζ_{1} : ⋯: ζ_{n}) ∈ ℙ^{n−1} where ζ_{1}, …, ζ_{n} are different nth roots of unity; this includes the lines that correspond to the points q_{ζ}. Now let Y′ be the blowup of Ỹ at the origin as before, and W be the blowup of Y′ at the lines that make up the strict transform of Z in Y′. (Alternatively, we may take the route similar to Remark 7.2 and set W to be the blowup of ℙ(Y) at the points q_{ζ}.) Then W does not have H-fixed points, and Proposition 7.1 shows that any x ∈ L_{n} satisfying (7.3) also satisfies the system (7.2), which in our case is

σ^{(1)}(x) = ⋯ = σ^{(n−1)}(x) = 0 .  (7.4)

One can now show directly that L_{n} does not have a non-zero element x satisfying (7.4); otherwise L_{n}/K_{n} would have to be a cyclic extension, a contradiction. Alternatively, one can show that the system (7.4) has no nontrivial solutions by applying Proposition 7.1 one more time, as follows:

— take the new H to be any cyclic subgroup of G = S_{n} of order different from n and 1;

— the new Y to be the old Z, i.e., P_{i} = s_{i}(x_{1}, …, x_{n}) for i = 1, …, n − 1.

— the new Z to be the origin in 𝔸^{n}, i.e., Q_{j} = s_{j}(x_{1}, …, x_{n}) for j = 1, …, n.

— the new W to be the normalization of Z, i.e., the disjoint union of (n − 1)! lines.

Applying Proposition 7.1 we see that the system (7.4) has no nontrivial solutions and, hence, neither does equation (7.3).

8. Equations in octonion algebras

Preliminaries. Let F be a field of characteristic ≠ 2. Recall that for any 0 ≠ a, b, c ∈ F, the octonion (or Cayley—Dickson) algebra O_{F} (a, b, c) is defined as follows. The quaternion algebra

(a, b)_{2} = F {i, j}/(i^{2} = a, j^{2} = b, ji = −ij)

is equipped with an involution x → \overline{x} given by

\overline{x_{0} + x_{1}i + x_{2}j + x_{3}ij} = x_{0} − x_{1}i − x_{2}j − x_{3}ij  (8.1)

for any x_{0}, …, x_{3} ∈ F. Now O_{F} (a, b, c) =^{def} (a, b)_{2} ⊕ (a, b)_{2}l is an 8-dimensional F - algebra with (non-associative) multiplication given by (x+yl)(z+wl) = (xz+c\overline{w}y)+ (wx + y\overline{z})l. The involution (8.1) extends from (a, b)_{2} to O_{F} (a, b, c) via \overline{x + yl} = \overline{x} − yl. The algebra O_{F} (a, b, c) is also equipped with F -valued trace and norm functions given by tr(x) = x+\overline{x} and n(x) = x\overline{x} = \overline{x}x such that x^{2}−tr(x)x+n(x) = 0 for any x ∈ O_{F} (a, b, c); we can think of tr(x) as σ^{(1)}(x) and n(x) as σ^{(2)}(x). Note that tr(x) is intrinsically defined in O_{K}(a, b, c), i.e., tr(x) = tr(σ(x)), where σ is a K-algebra automorphism in O_{K}(a, b, c); the same is true of n(x). For a more detailed description of octonion algebras we refer the reader to [Sc].

Two octonion algebras will be of particular interest to us: the split algebra O_{F} (1, 1, 1) over F and the generic algebra O_{gen} = O_{K}(a, b, c), where K = k(a, b, c) and a, b, c are algebraically independent over k.

By a theorem of Zorn [Sc, III.3.17], any 8-dimensional F -algebra A such that A ⊗_{F} F′ ≃ O_{F} (1, 1, 1) for some field extension F′/F, is necessarily isomorphic to O_{F} (a, b, c) for some a, b, c ∈ F^{∗}. This means that octonion algebras are “forms” of the split octonion algebra O_{k}(1, 1, 1) in the same way as central simple algebras are “forms” of the matrix algebra M_{n}(k).

G_{2}-equivariant maps. From now on we shall assume the base field k to be algebraically closed and of characteristic ≠ 2.

Recall that the automorphism group of the split octonion algebra O = O_{k}(1, 1, 1) is the exceptional group G_{2}. Octonion algebras are related to G_{2}-varieties in the same way as central simple algebras are related to PGL_{n}-varieties. In particular, if k is of characteristic 0 then any octonion algebra whose center is a finitely generated field extension of k can be written in the form RMaps_{G_{2}}(X, O), where O is viewed as an 8-dimensional vector space with the natural G_{2}-action and X is a generically free G_{2}-variety, uniquely determined up to birational isomorphism.

From now on, let H ≃ (ℤ/2)^{3} be the subgroup of G_{2} generated by τ_{1}, τ_{2} and τ_{3}, where

\begin{matrix}τ_{1}(i) = −i , & τ_{1}(j) = j , & τ_{1}(l) = l; \\ τ_{2}(i) = i , & τ_{2}(j) = −j , & τ_{2}(l) = l; \\ τ_{3}(i) = i , & τ_{3}(j) = j , & τ_{3}(l) = −l . \end{matrix}  (8.2)

Lemma 8.1. The generic octonion algebra O_{gen} is isomorphic to RMaps_{G_{2}}(X, V), where X = G_{2} ∗_{H} V and V = Span{i, j, k} is the 3-dimensional faithful representation of H given by (8.2).

Proof. The proof is similar to the proof of Lemma 4.2, so we will only outline it below.

Let α, β, γ be the coordinates of V relative to the basis {i, j, l}, let R = RMaps_{G_{2}}(X, O) and let π_{1}, π_{2}, π_{3} : X −→ O be the elements of R given by

\begin{matrix}π_{1} : [g, (α, β, γ)] ↦ αg(i) \\ π_{2} : [g, (α, β, γ)] ↦ βg(j) \\ π_{3} : [g, (α, β, γ)] ↦ γg(l) . \end{matrix}  (8.3)

It is easy to see that these maps are well-defined, i.e. π_{a}(g, v) = π_{a}(gh^{−1}, hv). Let

K = k(X)^{G_{2}} = k(V)^{H} = k(α^{2}, β^{2}, γ^{2}) .

We now identify O_{gen} with O_{K}(α^{2}, β^{2}, γ^{2}), and define ϕ: O_{gen} ⟶ R by ϕ(i) = π_{1}, ϕ(j) = π_{2} and ϕ(l) = π_{3}. Then ϕ is well-defined; see (8.3). Since O is a (nonassociative) division algebra, ϕ is injective. To see that ϕ is an isomorphism, we only need to show that dim_{K}(R) ≤ 8; this follows from [Re_{2}, Lemma 7.4(a)].

G_{2}-invariant polynomials. Consider the diagonal G_{2}-action on the 8mdimensional k-vector space W = O^{m}. Let P ∈ k[W]^{G_{2}} be a G-invariant polynomial and let A = O_{F} (a, b, c) be an octonion algebra. Identifying A with an F -subalgebra of A ⊗_{F} F′ ≃ O_{F′}(1, 1, 1), where F′ = F (\sqrt{a}, \sqrt{b}, \sqrt{c}), we can define P (a_{1}, …, a_{m}) for any a_{1}, …, a_{m} ∈ A. Arguing as in Lemma 2.3, we see that P (a_{1}, …, a_{m}) is well-defined and lies in F for any a_{1}, …, a_{m} ∈ A. (This also follows from a theorem of Schwarz [Sw, (3.23)], which asserts that k[W]^{G_{2}} is generated by elements of the form tr(M), where M is a monomial in u_{1}, …, u_{m} ∈ O.)

Proposition 8.2. Let H ≃ (ℤ/2)^{3} be the subgroup of G_{2} defined in (8.2). Suppose the subvariety Y of ℙ(O^{m}), cut out by homogeneous G_{2}-invariant polynomials P_{1} = ⋯ = P_{r} = 0, does not have an H-fixed point. Then the system

P_{1}(x_{1}, …, x_{m}) = ⋯ = P_{r}(x_{1}, …, x_{m}) = 0  (8.4)

has no non-trivial solutions in any octonion algebra of the form RMaps_{G_{2}}(X, O), where X is a G_{2}-variety with a smooth H-fixed point. In particular, the system (8.4) has no nontrivial solutions in the generic octonion algebra O_{gen}.

Proof. We argue as in the proof of Proposition 2.4. Assume, to the contrary, that (a_{1}, …, a_{m}) is a nontrivial solution of (8.4). Each a_{i} is a G_{2}-equivariant rational map X −→ O^{m}; together they define a G_{2}-equivariant rational map f: X −→ Y ⊂ ℙ(O^{m}). Applying the Going Down Theorem 2.1, we obtain a contradiction.

This proves the first assertion of the proposition. The second assertion follows from Lemma 8.1. Indeed, the variety X = G_{2} ∗_{H} V defined there has a smooth fixed point, namely (1, 0).

A system of equations. We are now ready to state and prove the main result of this section.

Theorem 8.3. Let Q(x_{1}, …, x_{m}) be (a non-commutative and non-associative) homogeneous polynomial of even degree in x_{1}, …, x_{m} such that Q(ϵ_{1}, …, ϵ_{m}) ≠ 0 for any (2s)-th roots of unity ϵ_{1}, …, ϵ_{m}, and let m and s be positive integers. Then the system

\{ \begin{matrix}tr(x_{1}^{2s}) = ⋯ = tr(x_{m}^{2s}) \\ tr(Q(x_{1}, …, x_{m})) = 0 . \end{matrix}  (8.5)

has no non-zero solutions in any octonion algebra of the form RMaps_{G_{2}}(X, O), where X is a generically free G_{2}-variety with a smooth H-fixed point. In particular, the system (8.5) has no nontrivial solutions in the generic octonion algebra O_{gen}.

Here H = <τ_{1}, τ_{2}, τ_{3}> ≃ (ℤ/2ℤ)^{3} is the subgroup of G_{2} defined in (8.2).

Proof. According to Proposition 8.2, it is enough to check that the variety

Y = { (U_{1} : ⋯: U_{m}) ∈ ℙ(O^{m}) | tr(U_{1}^{2s}) = ⋯ = tr(U_{m}^{2s}), tr(Q(U_{1}, …, U_{m})) = 0 }

(where U_{1}, …, U_{m} ∈ O are taken up to multiplication by an element of k) has no H-fixed points.

A point (U_{1} : ⋯: U_{m}) ∈ ℙ(O^{m}) is H-fixed iff all U_{r} lie in the same character space for the H-action on O. In other words, there exists a ζ ∈ {1, i, j, l, ij, il, jl, ijl} such that every U_{r} is of the form U_{r} = u_{r}ζ for some u_{r} ∈ k. Note that at least one u_{r} is non-zero; otherwise the point (U_{1} : ⋯: U_{m}) is not well-defined in ℙ(O^{m}). The condition that such a fixed point lies in Y translates into the system

\{ \begin{matrix}u_{1}^{2s} = ⋯ = u_{m}^{2s} \\ Q(u_{1}, …, u_{m}) = 0 \end{matrix}

of homogeneous equations in u_{1}, …, u_{m}. If u_{1} = 0 then the remaining u_{r} are also equal to 0, a contradiction. If u_{1} ≠ 0 then ϵ_{r} = u_{r}/u_{1} is a (2s)-th root of unity for each r = 1, …, m, and Q(ϵ_{1}, …, ϵ_{m}) = 0, contradicting our assumption on Q. This shows that Y has no H-fixed points.

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∗ Available at http://ucs.orst.edu/˜ reichstz/pub.html.

Department of Mathematics, Oregon State University, Corvallis, OR 97331

   E-mail address: zinovy@math.orst.edu

Department of Mathematics and Computer Science, University of the Negev, Be’er Sheva’, Israel

Current mailing address: Hashofar 26/3, Ma’ale Adumim, Israel

   E-mail address: youssin@math.bgu.ac.il