FIELDS OF DEFINITION FOR DIVISION ALGEBRAS

M. LORENZ, Z. REICHSTEIN, L. H. ROWEN, AND D. J. SALTMAN

Abstract. Let A be a finite-dimensional division algebra containing a base field k in its center F. We say that A is defined over a subfield F_{0} if there exists an F_{0}-algebra A_{0} such that A = A_{0} ⊗_{F_{0}} F. We show that: (i) In many cases A can be defined over a rational extension of k. (ii) If A has odd degree n ≥ 5, then A is defined over a field F_{0} of transcendence degree ≤ \frac{1}{2}(n − 1)(n − 2) over k. (iii) If A is a ℤ/m × ℤ/2-crossed product for some m ≥ 2 (and in particular, if A is any algebra of degree 4) then A is Brauer equivalent to a tensor product of two symbol algebras. Consequently, M_{m}(A) can be defined over a field F_{0} such that trdeg_{k}(F_{0}) ≤ 4. (iv) If A has degree 4 then the trace form of A can be defined over a field F_{0} of transcendence degree ≤ 4. (In (i), (iii) and (iv) we assume that the center of A contains certain roots of unity.)

Contents

1. Introduction 2 1.1. Parameter reduction 2 1.2. Rational fields of definition 3 1.3. ℤ/m × ℤ/2-crossed products 4 1.4. Fields of definition of quadratic forms 4 2. Preliminaries 5 2.1. G-lattices 5 2.2. Extension sequences 5 2.3. Twisted multiplicative G-fields 6 2.4. Rational specialization 7 3. G/H-crossed products 8 4. Proof of Theorem 1.1 12

1991 Mathematics Subject Classification. 16K20, 16W22, 20C10, 20J06, 11E81. Key words and phrases. division algebra, central simple algebra, crossed product, symbol

algebra, cyclic algebra, biquaternion algebra, Brauer factor set, Brauer group, integral representation, G-lattice, quadratic form, trace form, Pfister form, essential dimension.

M. Lorenz was supported in part by NSF grant DMS-9988756. Z. Reichstein was supported in part by NSF grant DMS-9801675 and an NSERC research

grant.

L. H. Rowen was supported by the Israel Science Foundation, founded by the Israel Academy of Sciences and Humanities - Center of Excellence Program no. 8007/99-3.

D. J. Saltman was supported in part by NSF grant DMS-9970213.

5. Proof of Theorem 1.2 14 5.1. Proof of part (a) 14 5.2. Proof of part(b) 16 6. Proof of Theorem 1.3 17 7. The field of definition of a quadratic form 19 7.1. Preliminaries 19 7.2. Proof of Theorem 1.5 21 References 22

1. Introduction

Let F be a field and A a finite-dimensional F -algebra. We say that A is defined over a subfield F_{0} ⊂ F if there exists an F_{0}-algebra A_{0} such that A = A_{0} ⊗_{F_{0}} F.

Throughout this paper we will assume that A is a finite-dimensional central simple F -algebra, and F (and F_{0}) contain a base field k.

1.1. Parameter reduction. If A is defined over F_{0} and trdeg_{k}(F_{0}) < trdeg_{k}(F) then the passage from A to A_{0} may be viewed as “parameter reduction” in A. This leads to the following natural question:

What is the smallest value of trdeg_{k}(F_{0}) such that A is defined over F_{0}?

This number is clearly finite; we shall denote it by τ(A). Of particular interest to us will be the case where A = UD(n) is the universal division algebra of degree n and F = Z(n) is the center of UD(n). Recall that UD(n) is the subalgebra of M_{n}(k(x_{ij}, y_{ij})) generated, as a k-division algebra, by two generic n × n-matrices X = (x_{ij}) and Y = (y_{ij}), where x_{ij} and y_{ij} are 2n^{2} independent commuting variables over k; see, e.g., [Pr, Section II.1], [Row_{1}, Section 3.2] or [Sa_{3}, Section 14]. We shall write d(n) for τ(UD(n)). It is easy to show that d(n) ≥ τ(A) for any central simple algebra A of degree n whose center contains k; see Remark 2.8 (cf. also [Re_{2}, Lemma 9.2]). In other words, every central simple algebra of degree n can be “reduced to at most d(n) parameters”.

To the best of our knowledge, the earliest attempt to determine the value of d(n) is due to Procesi, who showed that d(n) ≤ n^{2}; see [Pr, Thm. 2.1]. If n = 2, 3 or 6 and k contains a primitive n^{th} root of unity then d(n) = 2, because UD(n) is cyclic for these n and we can take A_{0} to be a symbol algebra; cf. [Re_{2}, Lemma 9.4]. Rost [Rost] recently proved that

d(4) = 5 .  (1.1)

For other n the exact value of d(n) is not known. However, the following inequalities hold:

d(n) ≤ n^{2} − 3n + 1 if n ≥ 4 [Le] ,
d(n) ≤ d(nm) ≤ d(n) + d(m) if (n, m) = 1 [Re_{2}, Sect. 9.4] ,
d(n^{r}) ≥ 2r [Re_{1}, Theorem 16.1] ,  (1.2)
d(n) ≤ \frac{1}{2}(n − 1)(n − 2) + n for odd n [Row_{2}]; cf. [Re_{2}, Sect. 9.3] .

In this paper we will sharpen the last inequality by showing that

d(n) ≤ \frac{1}{2}(n − 1)(n − 2)

for every odd n ≥ 5. Moreover, in UD(n), reduction to this number of parameters can be arranged in a particularly nice fashion: Theorem 1.1. Let n ≥ 5 be an odd integer, UD(n) the universal division algebra of degree n, and Z(n) its center. Then there exists a subfield F of Z(n) and a division algebra D of degree n with center F such that

(a) UD(n) = D ⊗_{F} Z(n),

(b) trdeg_{k}(F) = \frac{1}{2}(n − 1)(n − 2), and

(c) Z(n) is a rational extension of F. In particular, d(n) ≤ \frac{1}{2}(n − 1)(n − 2).

We remark that in the language of [Re_{2}], the last assertion of Theorem 1.1 can be written as

ed(PGL_{n}) ≤ \frac{1}{2}(n − 1)(n − 2).  (1.3)

1.2. Rational fields of definition. Another natural question is whether or not a given central simple algebra A can be defined over a rational extension of k. We give the following partial answer to this question. Theorem 1.2. Let A be a finite-dimensional central simple algebra with center F and let t_{1}, t_{2}, … be algebraically independent central indeterminates over F.

(a) Assume deg(A) = 2^{i}p_{1} … p_{r}, where i = 0, 1 or 2 and p_{1}, …, p_{r} are distinct odd primes. Then for s ≫ 0 the algebra A(t_{1}, …, t_{s}) is defined over a rational extension of k.

(b) Suppose the center of A contains a primitive e^{th} root of unity, where e is the exponent of A. Then there exists an r ≥ 1 such that for s ≫ 0 the algebra M_{r}(A)(t_{1}, …, t_{s}) is defined over a rational extension of k. (Here we are imposing no restrictions on the degree of A.)

Here A(t_{1}, …, t_{s}) stands for A⊗_{F} F (t_{1}, …, t_{s}) and similarly for M_{r}(A)(t_{1}, …, t_{s}). Note that part (b) may be interpreted as saying that for s ≫ 0 the Brauer class of A(t_{1}, …, t_{s}) is defined over a rational extension of k.

1.3. ℤ/m × ℤ/2-crossed products. As usual, we let (a, b)_{m} denote the symbol algebra

F {x, y}/(x^{m} = a, y^{m} = b, xy = ζ_{m}yx) ,  (1.4)

where a, b ∈ F^{∗} and ζ_{m} is a (fixed) primitive m^{th} root of unity in F; cf. [Row_{3}, pp. 194-197]. In Section 6 we will prove the following: Theorem 1.3. Let A be a ℤ/m × ℤ/2-crossed product central simple algebra whose center F contains a primitive 2m^{th} root of unity ζ_{2m}. Then A is Brauer equivalent (over F) to (a, b)_{m} ⊗_{F} (c, d)_{2m} for some a, b, c, d ∈ F^{∗}. (In other words, M_{m}(A) is isomorphic to (a, b)_{m} ⊗_{F} (c, d)_{2m}.)

Note that Theorem 1.3 applies to any division algebra of degree 4, since, by a theorem of Albert, any such algebra is a ℤ/2 × ℤ/2-crossed product. In this setting our argument yields, in particular, an elementary proof of [S, Theorem 2, p. 288]. We also remark that Theorem 1.3 may be viewed as an explicit form of the Merkurjev-Suslin theorem for ℤ/m × ℤ/2-crossed products.

Letting F_{0} = k(ζ_{2m}, a, b, c, d), we note that

(a, b)_{m} ⊗_{F} (c, d)_{2m} = ((a, b)_{m} ⊗_{F_{0}} (c, d)_{2m}) ⊗_{F_{0}} F .

Thus Theorem 1.3 shows that M_{m}(A) is defined over F_{0}. Since trdeg_{k}(F_{0}) ≤ 4, we obtain the following: Corollary 1.4. Let A be a ℤ/m × ℤ/2-crossed product central simple algebra whose center contains a primitive 2m^{th} root of unity. Then τ(M_{m}(A)) ≤ 4. In particular, τ(M_{2}(A)) ≤ 4 for every central simple algebra A of degree 4 whose center contains a primitive 4^{th} root of unity. □

Note that the last assertion complements, in a somewhat surprising way, the above-mentioned result of Rost (1.1). Indeed, suppose the base field k contains a primitive 4^{th} root of unity. Then for A = UD(4), Rost’s theorem says that τ(A) = 5, where as Corollary 1.4 says that τ(M_{2}(A)) ≤ 4.

1.4. Fields of definition of quadratic forms. A quadratic form q: V → F on an F -vector space V = F^{n} is said to be defined over a subfield F_{0} of F if q = q_{F_{0}} ⊗ F, where q_{F_{0}} is a quadratic form on V_{0} = F_{0}^{n}.

In the last section we discuss fields of definition of quadratic forms. Of particular interest to us will be trace forms of central simple algebras of degree 4, recently studied by Rost, Serre and Tignol [RST]. (Recall that the trace form of a central simple algebra A is the quadratic form x ↦ Tr(x^{2}) defined over the center of A.) We will use a theorem of Serre [Se] (see our Proposition 7.3) to prove the following: Theorem 1.5. Let A be a central simple algebra of degree 4 whose center F contains a primitive 4^{th} root of unity. Then the trace form of A is defined over a subfield F_{0} ⊂ F such that trdeg_{k}(F_{0}) ≤ 4.

Note that Theorem 1.5 may also be viewed as complementing (1.1).

Acknowledgment. The authors would like to thank the referee for a number of helpful and constructive comments and for catching several mistakes in an earlier version of this paper.

2. Preliminaries

2.1. G-lattices. Throughout, G will denote a finite group and H will be a subgroup of G. Recall that a G-module is a (left) module over the integral group ring ℤ[G]. As usual, Ext_{G} stands for Ext_{ℤ[G]}. A G-lattice is a G-module that is free of finite rank over ℤ. Further, a G-lattice M is called

• a permutation lattice if M has a ℤ-basis that is permuted by G;

• permutation projective (or invertible) if M is a direct summand of some

permutation G-lattice. A G-module M is called faithful if no 1 ≠ g ∈ G acts as the identity on M.

The G/H-augmentation kernel is defined as the kernel of the natural augmentation map

ℤ[G/H] = ℤ[G] ⊗_{ℤ[H]} ℤ ⟶ ℤ , \overline{g} = g ⊗ 1 ↦ 1 (g ∈ G) .

Thus there is a short exact sequence of G-lattices

0 → ω(G/H) → ℤ[G/H] → ℤ → 0 .  (2.1)

ω(G/{1}) will be written as ωG; this is the ordinary augmentation ideal of the group ring ℤ[G]; cf. [Pa, Chap. 3].

2.2. Extension sequences. Exact sequences of G-lattices of the form

0 → M → P → ω(G/H) → 0 ,

with P permutation and M faithful

(2.2)

will play an important role in the sequel. In this subsection we introduce two such sequences, (2.3) and (2.4).

Let d_{G}(ω(G/H)) denote the minimum number of generators of ω(G/H) as a ℤ[G]-module. Then for any r ≥ d_{G}(ω(G/H)) there exists an exact sequence

0 → M → ℤ[G]^{r} →^{f} ω(G/H) → 0 .  (2.3)

of G-lattices. Lemma 2.1. M is a faithful G-lattice if and only if r ≥ 2 or H ≠ {1}.

Proof. It is enough to show that M ⊗ ℚ is G-faithful; thus we may work over the semisimple algebra ℚ[G]. Since f ⊗ ℚ splits, we have a ℚ[G]-isomorphism (ω(G/H) ⊗ ℚ) ⊕ (M ⊗ ℚ) ≃ ℚ[G]^{r}. Similarly, the canonical exact sequence ℤ[G]ωH ↣ ℤ[G] ↠ ℤ[G/H] gives (ω(G/H) ⊗ ℚ) ⊕ ℚ ⊕ ℚ[G]ωH ≃ ℚ[G]. Therefore,

M ⊗ ℚ ≃ ℚ[G]^{r−1} ⊕ ℚ ⊕ ℚ[G]ωH .

If r ≥ 2 then ℚ[G]^{r−1} is G-faithful, and if H ≠ {1} then ωH ⊗ ℚ is H-faithful and so ℚ[G]ωH ≃ (ωH ⊗ ℚ)↑_{H}^{G} is G-faithful. In either case, M ⊗ ℚ is faithful, as desired. On the other hand, r = 1 and H = {1} leads to M ⊗ ℚ ≃ ℚ which is not faithful. □

Lemma 2.2. There is an exact sequence

0 → ω(G/H)^{⊗2} →^{m} P → ω(G/H) → 0 ,  (2.4)

where P is the (permutation) sublattice P = ⨁_{\overline{g_{1}}≠\overline{g_{2}}∈G/H} ℤ(\overline{g_{1}} ⊗\overline{g_{2}}) of ℤ[G/H]^{⊗2}. The lattice ω(G/H)^{⊗2} is faithful if and only if H contains no normal subgroup ≠ {1} of G and [G : H] ≥ 3.

Proof. Tensoring sequence (2.1) with ω(G/H) and putting P′ = ω(G/H) ⊗ ℤ[G/H], we obtain an exact sequence

0 → ω(G/H)^{⊗2} → P′ → ω(G/H) → 0 ,

where ⊗ = ⊗_{ℤ}. The elements (\overline{g_{1}} − \overline{g_{2}}) ⊗ \overline{g_{2}} with \overline{g_{1}} ≠ \overline{g_{2}} ∈ G/H form a ℤ-basis of P′, and the map

m: (\overline{g_{1}} − \overline{g_{2}}) ⊗ \overline{g_{2}} ↦ \overline{g_{1}} ⊗ \overline{g_{2}}

is a G-isomorphism P′ ≃ P.

For the faithfulness assertion, note that N = ⋂_{g∈G} H^{g} acts trivially on ℤ[G/H] and hence on ω(G/H)^{⊗2}; so N = {1} is surely required for faithfulness. Also, if [G : H] ≤ 2 then G acts trivially on ω(G/H)^{⊗2}. Conversely, if N = {1} and [G : H] ≥ 3 then it is easy to verify that ω(G/H)^{⊗2} is indeed faithful. □

2.3. Twisted multiplicative G-fields. Recall that a G-field is a field F on which the finite group G acts by automorphisms, written f ↦ g(f). Morphisms of G-fields are G-equivariant field homomorphisms. The G-field F is called faithful if every 1 ≠ g ∈ G acts non-trivially on F. If K ⊆ F is a field extension and V ⊆ F a subset of F (not necessarily algebraically independent over K) then we let K(V) denote the subfield of F that is generated by K and V. Lemma 2.3. (cf. [Sh, Appendix 3]) Let K ⊆ F be an extension of G-fields with K faithful. Assume that F = K(V) for some G-stable K-subspace V ⊆ F. Then

(a) V = KV^{G}, where V^{G} denotes the G-invariants in V,

(b) F = K(V^{G}), and

(c) F^{G} = K^{G}(V^{G}).

Proof. (a) Let S = K#G denote the skew group ring for the given G-action on K. The G-action on F and multiplication with K make F a (left) S-module, and V is a submodule. Moreover, since K is a faithful G-field, S is a simple ring; see, e.g., [J, p. 473]. In particular, the element t = ∑_{g∈G} g ∈ S generates S as a 2-sided ideal. Thus, S = StS = KtK and consequently, V = KtKV = KV^{G}.

(b) is an immediate consequence of (a).

(c) Let E = K^{G}(V^{G}). We want to show that E = F^{G}. Clearly E ⊆ F^{G}. To prove equality, note that KE is a subring of F containing K and V^{G}, and that dim_{E} KE ≤ dim_{K^{G}} K = |G|. Thus, KE is a field, and hence (b) implies that KE = F. Therefore, dim_{E} F ≤ |G| = dim_{F^{G}} F. Since E ⊆ F^{G}, this is only possible if E = F^{G}. □

We recall a well-known construction of G-fields; cf. [Sa_{1}]. Given a G-field E, a G-lattice M, and an extension class γ ∈ Ext_{G}(M, E^{∗}), the twisted multiplicative G-field E_{γ}(M) is constructed as follows. Form the group algebra E[M] of M over E; this is a commutative integral domain with group of units U(E[M]) = E^{∗}×M. We shall use multiplicative notation for M in this setting. Let E(M) denote the field of fractions of E[M]. Choose an extension of G-modules

1 → E^{∗} → V → M → 1  (2.5)

representing γ. So, as abelian groups, V ≃ U(E[M]). Using this identification, we obtain a G-action on U(E[M]) inducing the given action on E^{∗}. The action of G on U(E[M]) extends uniquely to E[M], and to E(M); we will use E_{γ}[M] and E_{γ}(M) to denote E[M] and E(M) with the G-actions thus obtained. For γ = 1, we will simply write E[M] and E(M). We remark that the choice of the sequence (2.5) representing a given γ ∈ Ext_{G}(M, E^{∗}) is insubstantial: a different choice leads to G-isomorphic results.

For future reference, we record the following application of Lemma 2.3 essentially due to Masuda [Mas]; see also [Le, Proposition 1.6], [Sa_{3}, Lemma 12.8]. Proposition 2.4. Let E be a faithful G-field and let P be a permutation G-lattice. Then any twisted multiplicative G-field E_{γ}(P) can be written as

E_{γ}(P) = E(t_{1}, …, t_{n})

with G-invariant transcendental (over E) elements t_{i}. In particular, E_{γ}(P)^{G} = E^{G}(t_{1}, …, t_{n}) is rational over E^{G}.

Proof. We have an extension of G-modules 1 → E^{∗} → U(E_{γ}[P]) → P → 1 representing γ, as in (2.5). Fix a ℤ-basis, X, of P that is permuted by the action of G. For each x ∈ X, choose a preimage x′ ∈ U(E_{γ}[P]). Then {x′}_{x∈X} is a collection of transcendental generators of E_{γ}(P) over E, and G acts via g(x′) = g(x)′y for some y = y(g, x) ∈ E^{∗}. Letting V denote the E-subspace of E_{γ}(P) that is generated by {x′}_{x∈X}, we conclude from Lemma 2.3 that V has a basis consisting of G-invariant elements, say t_{1}, …, t_{n}, and E_{γ}(P) = E(t_{1}, …, t_{n}), E_{γ}(P)^{G} = E^{G}(t_{1}, …, t_{n}). The t_{i} are transcendental over E, since trdeg_{E} E_{γ}(P) = rank(P) = n. □

2.4. Rational specialization. Let A/F and B/K be central simple algebras. We will call B/K a rational specialization of A/F if there exists a field F′ containing both F and K such that F′/K is rational and

B ⊗_{K} F′ ≃ A ⊗_{F} F′ .

In other words, B is a rational specialization of A if deg A = deg B and A embeds in some B(t_{1}, …, t_{n}), where t_{1}, t_{2}, … are independent variables over F; cf. [Sa_{3}, p. 73],

For the rest of this paper we fix an (arbitrary) base field k. All other fields are understood to contain a copy k and all maps (i.e., inclusions) between fields are understood to restrict to the identity map on k. Definition 2.5. Let Λ be a class of central simple algebras. We shall say that an algebra A ∈ Λ has the rational specialization property in the class Λ if every B ∈ Λ is a rational specialization of A. If Λ is the class of all central simple algebras of degree n = deg(A) then we will omit the reference to Λ and will simply say that A has the rational specialization property. Example 2.6. By [RV, Lemma 3.1], UD(n) has the rational specialization property. This is also implicit in [Sa_{2}]. We remark that any central simple algebra A/F with the rational specialization property is a division algebra. To see this, specialize A to UD(n), where n = deg(A).

Recall the definition of τ(A) given at the beginning of this paper. Lemma 2.7. Let A/F and B/K be a central simple algebras.

(a) If A′ ≃ A⊗_{F} F′ for some rational field extension F′/F then τ(A) = τ(A′).

(b) (cf. [Sa_{3}, Lemma 11.1]) If A is a rational specialization of B then τ(A) ≤ τ(B).

Proof. (a) The inequality τ(A′) ≤ τ(A) is immediate from the definition of τ. To prove the opposite inequality, suppose A′ ≃ A_{0} ⊗_{F_{0}} F′, where A_{0} is a central simple algebra over an intermediate field k ⊂ F_{0} ⊂ F′, and A_{0} is chosen so that trdeg_{k}(F_{0}) = τ(A′). In particular, trdeg_{k}(F_{0}) ≤ trdeg_{k}(F). Then by [RV, Proposition 3.2], A_{0} embeds in A, i.e., A ≃ A_{0} ⊗_{F_{0}} F for some embedding F_{0} ↪ F. Consequently, τ(A) ≤ trdeg_{k}(K) = τ(A′), as desired.

(b) We may assume B ⊗_{K} F′ = A′, as in (a). Clearly τ(B) ≥ τ(A′), and part (a) tells us that τ(A′) = τ(A). □

Remark 2.8. Combining Example 2.6 with Lemma 2.7(b), we see that τ(A) ≤ d(n) = τ(UD(n)) holds for every central simple algebra A of degree n; cf. [Re_{2}, Lemma 9.2].

3. G/H-crossed products

We shall call a central simple algebra A/F an (E, G/H)-crossed product if A has a maximal subfield L whose Galois closure E over F has the property that Gal(E/F) = G and Gal(E/L) = H. (We adopt the convention that a maximal subfield of A is a subfield L that is maximal as a commutative subring; so [L : F] is equal to the degree of A.) We will say that A is a G/H-crossed product if it is an (E, G/H)-crossed product for some faithful G-field E. If H = {1} then a G/H-crossed product is just a G-crossed product in the usual sense (see, e.g., [Row_{1}, Definition 3.1.23]). Example 3.1. Consider the universal division algebra UD(n) generated by two generic matrices, X and Y, over k. Denote the center of this algebra by Z(n). Setting L = Z(n)(X), we see that UD(n) is an S_{n}/S_{n−1}-crossed product [Pr, Theorem 1.9]; see also Section 4 below.

Since the degree of a G/H-crossed product is equal to [G : H], we see that isomorphism classes of (E, G/H)-crossed products are in 1-1 correspondence with the relative Brauer group B(L/F), which, in turn, is naturally identified with the kernel of the restriction homomorphism H^{2}(G, E^{∗}) → H^{2}(H, E^{∗}); cf. [Pi, 14.7].

A G-module M is called H^{1}-trivial if H^{1}(H, M) = 0 holds for every subgroup H ≤ G. Equivalently, M is H^{1}-trivial if Ext_{G}(P, M) = 0 for all permutation projective G-lattices P; see, e.g., [Sa_{3}, Lemma 12.3]. Lemma 3.2. Given an exact sequence

0 → M → P → ω(G/H) → 0 ,

of G-lattices, with P -permutation, let N be an H^{1}-trivial G-module. Denote the kernel of the restriction homomorphism H^{2}(G, N) → H^{2}(H, N) by K(G/H, N). Then there is a natural isomorphism

ϕ_{N}: Hom_{G}(M, N)/ Im(Hom_{G}(P, N)) ⟶^{≃} K(G/H, N) .

Here “natural” means that for every homomorphism N → N′ of H^{1}-trivial G-modules, the following diagram commutes

\xymatrix{Hom_{G}(M, N′)/ Im(Hom_{G}(P, N′)) \ar[r]^{ϕ_{N′}} & K(G/H, N′) \\ Hom_{G}(M, N)/ Im(Hom_{G}(P, N)) \ar[u] \ar[r]^{ϕ_{N}} & K(G/H, N) \ar[u]}  (3.1)

Note that, other than in sequence (2.2), the G-lattice M need not be faithful.

Proof. The lemma is a variant of [Sa_{3}, Theorem 12.10], where the same assertion is made for the sequence (2.4). The proof of [Sa_{3}, Theorem 12.10] goes through unchanged in our setting. □

In subsequent applications we will always take N = E^{∗}, where E is a faithful G-field. Note that E^{∗} is H^{1}-trivial by Hilbert’s Theorem 90. As we remarked above, K(G/H, E^{∗}) is naturally identified with B(L/F), where L = E^{H}, and elements of B(L/F) are in 1-1 correspondence with (E, G/H)-crossed products.

We shall denote the (E, G/H)-crossed product associated to a G-homomorphism f: M → E^{∗} by Alg(f). Lemma 3.3. Consider a sequence of G-lattices of the form (2.2). Let E be Gfield and f: M → E^{∗} be a homomorphism of G-modules. If k(f(M)) is contained in a faithful G-subfield E_{0} of E then Alg(f) is defined over E_{0}^{G}.

Proof. Since f is the composition of f_{0} : M → E_{0}^{∗} with the inclusion E_{0}^{∗} ↪ E^{∗}, Lemma 3.2 tells us that A = Alg(f_{0}) ⊗_{E_{0}^{G}} E^{G}. □

Remark 3.4. In the special case where the sequence

0 → M → P → ω(G/H) → 0

is given by (2.4), M = ω(G/H)^{⊗2} has a particularly convenient set of generators

y_{ijh} = (\overline{g_{i}} − \overline{g_{j}}) ⊗ (\overline{g_{j}} − \overline{g_{h}}) ,

where G/H = \{\overline{g_{1}}, …, \overline{g_{n}}\} is the set of left cosets of H in G and i, j, h range from 1 to n = [G : H]; cf. [RS, Lemma 1.2]. If f: ω(G/H)^{⊗2} → E^{∗} is a G-module homomorphism then the elements c_{ijh} = f(y_{ijh}) form a reduced Brauer factor set for Alg(f) in the sense of [RS, p. 449]. Conversely, for any reduced Brauer factor set (c_{ijh}) in E^{∗}, there exists a homomorphism f: ω(G/H)^{⊗2} → E^{∗} such that f(y_{ijh}) = c_{ijh}; see [RS, Corollary 1.3]. Thus Lemma 3.3 takes the following form:

Let A be an (E, G/H)-crossed product defined by a reduced Brauer factor set (c_{ijh}). Suppose (c_{ijh}) is contained in a faithful G-subfield E_{0} of E. Then A is defined over E_{0}^{G}. □

This following theorem is a variant of [Sa_{3}, Theorem 12.11]. Theorem 3.5. Given the sequence (2.2), let μ : M ↪ k(M)^{∗} be the natural inclusion. Then D = Alg(μ) has the rational specialization property in the class of G/H-crossed products containing a copy of k in their center. In particular, τ(A) ≤ rank(M) holds for any G/H-crossed product A/F with k ⊂ F.

Proof. Write A = Alg(f) for some G-homomorphism f: M → E^{∗}, where E is a faithful G-field with E^{G} = F; see the remarks following Lemma 3.2. Furthermore, let E(P) denote the fraction field of the group algebra E[P], with the G-action induced from the G-actions on E and P. By Proposition 2.4, there exists an Eisomorphism j: E(P) ≃ E(t) of G-fields, where t = (t_{1}, …, t_{r}), r = rank(P), are indeterminates on which G acts trivially. Therefore, E(P)^{G} ≃ E^{G}(t) = F (t) is a rational extension of F. Let f_{t} : M → E(t)^{∗} denote the composition of f with the natural inclusion E^{∗} ↪ E(t)^{∗}. Then Alg(f_{t}) = Alg(f) ⊗_{F} F (t) = A ⊗_{F} F (t). By Lemma 3.2, Alg(f_{t}) ≃ Alg(f_{t} + g|_{M}) for any g ∈ Hom_{G}(P, E(t)^{∗}). Let g be the composite g: P ↪ E(P)^{∗} ⟶^{∼} E(t)^{∗} and let φ be the G-module map φ: M → E(t)^{∗}, φ(m) = f_{t}(m)g(m). We claim that φ lifts to an embedding of G-fields k(M) ↪ E(t). Indeed, modulo E^{∗}, φ(m) ≡ g(m) ∈ P ⊆ E(t)^{∗}.

Hence, {φ(m)}_{m∈M} is an E-linearly independent subset of E(t), and so the map k[φ]: k[M] → E(t), ∑_{m} k_{m}m ↦ ∑_{m} k_{m}φ(m) is a G-equivariant embedding of the group ring k[M] into E(t). This embedding lifts to an embedding of G-fields ϕ: k(M) = Q(k[M]) ↪ E(t), as we have claimed. So ϕ ∘ μ = φ, and hence D ⊗_{k(M)^{G}} F (t) = Alg(ϕ ∘ μ) = Alg(φ) ≃ Alg(f_{t}) = A ⊗_{F} F (t). This proves that A is a rational specialization of D.

Lemmas 2.7(b) and 3.3 now imply that τ(A) ≤ τ(D) ≤ trdeg_{k} k(M)^{G} = rank(M).

This completes the proof of the theorem. □

Remark 3.6. Continuing with the notation used in the above theorem, the rational specialization property of D = Alg(μ) implies that D is a division algebra of exponent [G : H]. Indeed, by [FSS, Appendix] there exists a G/Hcrossed product division algebra of exponent [G : H], and the above assertion can be proved by specializing D to this algebra. Alternatively, the fact that D is a division algebra of exponent [G : H] can be checked directly by showing that the image of μ in H^{2}(G, k(M)^{∗}) (see Lemma 3.2) has order [G : H]. Remark 3.7. The above construction applies in particular to sequences of the form (2.3). The following special type of sequence (2.3) has been particularly well-explored. Write G = ⟨H, g_{1}, …, g_{r}⟩ for suitable g_{i} ∈ G; the minimal such r is usually denoted by d(G/H). Then we can define an epimorphism of Glattices f: ℤ[G]^{r} ↠ ω(G/H), f(α_{1}, …, α_{r}) = ∑_{i=1}^{r} α_{i}\overline{(g_{i} − 1)}, where : ℤ[G] ↠ ℤ[G/H] is the canonical map; see [Pa, Lemma 3.1.1]. The kernel R(G/H) = Ker f is called a relative relation module; it has the following group theoretical description. Let F_{r} denote the free group on r generators and consider the presentation

1 → R → F_{r} ∗ H → G → 1  (3.2)

where F_{r} ∗ H → G is the identity on H and sends the r generators of F_{r} to the elements g_{1}, …, g_{r}. Then R(G/H) ≃ R^{ab} = R/[R, R], with G acting by conjugation; see [Ki] and [Gr] (for H = {1}). Thus, we have the following version of sequence (2.3) with M = R^{ab} :

0 → R^{ab} → ℤ[G]^{r} → ω(G/H) → 0 .  (3.3)

When H = {1} and r ≥ 2, the division algebra D constructed via (3.3) in Theorem 3.5 is identical with the generic G-crossed product of Snider [S]; see also Rosset [Ro]. Explicitly:

Given a free presentation 1 → R → F_{r} → G → 1 of G with r ≥ 2, let M = R^{ab} ≤ \overline{F_{r}} = F_{r}/[R, R] and let a ∈ H^{2}(G, k(M)^{∗}) be the image of the extension class [1 → M → \overline{F_{r}} → G → 1] ∈ H^{2}(G, M) under the natural inclusion μ : M ↪ k(M)^{∗}. Then D = Alg(μ) is the G-crossed product (k(M), G, a) or, equivalently, the localization of the group algebra k[\overline{F_{r}}] at the nonzero elements of k[M].

Corollary 3.8. Let A be a G/H-crossed product and let d_{G}(ω(G/H)) be the minimal number of generators of ω(G/H) as a G-module. Then

τ(A) ≤ r|G| − [G : H] + 1 ,

where

r = \begin{cases}d_{G}(ω(G/H)) & if H ≠ \{1\} \\ max\{2, d_{G}(ω(G/H))\} & if H = \{1\} \end{cases}

Proof. Applying Theorem 3.5 to the exact sequence (2.3), we obtain

τ(A) ≤ rank(M) = rank(ℤ[G]^{r}) − rank(ω(G/H)) = r|G| − [G : H] + 1 ,

as claimed. Note that for r as above, Lemma 2.1 tells us that M is faithful, so that Theorem 3.5 is, indeed, applicable. □

Remark 3.9. As we pointed out in Remark 3.7, d_{G}(ω(G/H)) ≤ d(G/H). The difference pr(G/H) = d(G/H) − d_{G}(ω(G/H)) ≥ 0 can be arbitrarily large, even if H = {1}. In this case d(G) = d(G/{1}) is the minimal number of generators of G, and pr(G) = d(G) − d_{G}(ωG) is usually called the presentation rank or generation gap of G. All solvable groups G have presentation rank pr(G) = 0; see [Gr, Lectures 6 and 7]. Moreover, if the derived subgroup [G, G] is nilpotent then pr(G/H) = 0 holds for every subgroup H of G; see [Ki]. Corollary 3.10. (a) Suppose a group G of order n can be generated by r ≥ 2

elements. Then τ(A) ≤ (r − 1)n + 1 for any G-crossed product central simple algebra A.

(b) τ(A) ≤ (⌊log_{2}(n)⌋ − 1)n + 1 holds for any crossed product central simple algebra A of degree n ≥ 4.

Here, as usual, ⌊x⌋ denotes the largest integer ≤ x.

Proof. (a) is an immediate consequence Corollary 3.8. (b) follows from (a), because any group of order n can be generated by r ≤ log_{2}(n) elements. (Indeed, |⟨G_{0}, g⟩| ≥ 2|G_{0}| for any subgroup G_{0} of G and any g ∈ G \ G_{0}.) Note also that ⌊log_{2}(n)⌋ ≥ 2 for any n ≥ 4. □

4. Proof of Theorem 1.1

For the next two sections we shall assume that G = S_{n} and H = S_{n−1}. We will use the following standard notations for S_{n}-lattices:

ℤ[S_{n}/S_{n−1}] = U_{n} and ω(S_{n}/S_{n−1}) = A_{n−1} .  (4.1)

The natural generators of U_{n} will be denoted by u_{1}, …, u_{n}; the symmetric group S_{n} permutes them via σ(u_{i}) = u_{σ(i)}. A_{n−1} is the sublattice of U_{n} generated by u_{i} − u_{1} as i ranges from 2 to n.

Recall that the universal division algebra UD(n) is generated, as a k-division algebra, by a pair of generic n × n-matrices X and Y. We may assume without loss of generality that X is diagonal. Following [Row_{2}] we will denote the diagonal entries of X by ζ′_{ii} and the entries of Y by ζ_{ij}, where ζ′_{ii} and ζ_{ij} are algebraically independent variables over k. The group S_{n} permutes these variables as follows:

σ(ζ′_{ii}) = ζ′_{σ(i)σ(i)} and σ(ζ_{ij}) = ζ_{σ(i)σ(j)}.

We identify the multiplicative group generated by ζ′_{ii} with the S_{n}-lattice U_{n} (via ζ′_{ii} ↔ u_{i}), and the multiplicative group generated by ζ_{ij} with U_{n} ⊗ U_{n} (via ζ_{ij} ↔ u_{i} ⊗ u_{j}). Consider the exact sequence

0 → Ker(f) → U_{n} ⊕ U_{n}^{⊗2} →^{f} A_{n−1} → 0  (4.2)

of S_{n}-lattices, where f(u_{i}, u_{j} ⊗u_{h}) = u_{j} −u_{h}. This sequence is the sequence (2.4) of Lemma 2.2 for G = S_{n} and H = S_{n−1}, with two extra copies of U_{n} added: the second copy of U_{n} is the sublattice of U_{n}^{⊗2} that is spanned by all elements u_{i} ⊗ u_{i}. Both copies of U_{n} belong to Ker(f); in fact,

Ker(f) = U_{n} ⊕ U_{n} ⊕ A_{n−1}^{⊗2} ,

where A_{n−1}^{⊗2} is identified with the sublattice of U_{n}^{⊗2} that is spanned by all elements (u_{i} − u_{j}) ⊗ (u_{l} − u_{m}).

Let E = k(Ker(f)) and F = E^{S^{n}}. By a theorem of Formanek and Procesi, F is naturally isomorphic to the center Z(n) of UD(n); see, e.g., [F_{1}, Theorem 3]. Note that E = F (ζ′_{11}, …, ζ′_{nn}) is generated over F by the eigenvalues of the generic matrix X. Consequently, UD(n) is an (E, S_{n}/S_{n−1})-product, and E^{S^{n−1}} is isomorphic to the maximal subfield Z(n)(X) of UD(n); see, [Pr, Section II.1].

Theorem 1.1 is now a consequence of the following: Proposition 4.1. Suppose n ≥ 5 is odd. Then

(a) UD(n) is defined over F_{0} = k(⋀^{2} A_{n−1})^{S^{n}},

(b) Z(n) = k(Ker(f))^{S^{n}} is rational over F_{0} = k(⋀^{2} A_{n−1})^{S^{n}}, Here, we view ⋀^{2} A_{n−1} as the sublattice of antisymmetric tensors in A_{n−1}^{⊗2},

that is, the ℤ-span of all a ∧ a′ = a ⊗ a′ − a′ ⊗ a with a, a′ ∈ A_{n−1}.

Proof. We will deduce part (a) from Remark 3.4 by constructing a reduced Brauer factor set contained in E_{0} = k(⋀^{2} A_{n−1}). First we note that the S_{n}- action on E_{0} is faithful, because ⋀^{2} A_{n−1} is a faithful S_{n}-lattice for every n ≥ 4. (Indeed, ⋀^{2} A_{n−1} ⊗ ℚ is the simple S_{n}-representation corresponding to the partition (n − 2, 1^{2}) of n; cf. [FH, Exercise 4.6]).

We now proceed with the construction of the desired Brauer factor set. The computation in [Row_{2}, Section 2] shows that the elements

c_{ijh} = ζ_{ij}ζ_{jh}ζ_{ih}^{−1} ∈ E^{∗} .

form a Brauer factor set for UD(n). By [Row_{2}, Theorem 4], if n is odd, UD(n) has a normalized (and, in particular, reduced) Brauer factor set (c′_{ijh}) given by

c′_{ijh} = (c_{ijh}/c_{hji})^{\frac{n+1}{2}} = (ζ_{ij}ζ_{ji}^{−1}ζ_{jh}ζ_{hj}^{−1}ζ_{hi}ζ_{ih}^{−1})^{\frac{n+1}{2}} .

Now observe that ζ_{ij}ζ_{ji}^{−1}ζ_{jh}ζ_{hj}^{−1}ζ_{hi}ζ_{ih}^{−1} is precisely the element of U_{n}^{⊗2} we identified with (u_{i} − u_{j}) ∧ (u_{j} − u_{h}). Thus every c′_{ijh} lies in ⋀^{2} A_{n−1} ⊂ E_{0}, as desired.

(b) The canonical exact sequence

0 → ⋀^{2} A_{n−1} ⟶ A_{n−1}^{⊗2} ⟶ Sym^{2} A_{n−1} → 0

of S_{n}-lattices gives rise to an exact sequence

0 → ⋀^{2} A_{n−1} → A_{n−1}^{⊗2} ⊕ U_{n} ⊕ ℤ → Q → 0 ,

where we have put Q = Sym^{2} A_{n−1}⊕U_{n}⊕ℤ. The crucial fact here is that, by [LL, Section 3.5], if n is odd, Q is a permutation lattice. Applying Proposition 2.4 to the extension of (faithful) S_{n}-fields E_{0} = k(⋀^{2} A_{n−1}) ⊆ k(A_{n−1}^{⊗2} ⊕ U_{n} ⊕ ℤ) ≅ (E_{0})_{γ}(Q), where γ is the image of the class of the above extension in Ext_{G}(Q, E_{0}^{∗}), we conclude that

k(A_{n−1}^{⊗2} ⊕ U_{n} ⊕ ℤ) ≃ k(⋀^{2} A_{n−1})(x_{1}, …, x_{m})

as S_{n}-fields, where m = \frac{n(n+1)}{2} + 1 and S_{n} acts trivially on the x_{i}’s. Similarly, putting L_{n} = A_{n−1}^{⊗2} ⊕ U_{n}^{2}, the obvious sequence 0 → A_{n−1}^{⊗2} ⊕ U_{n} → L_{n} → U_{n} → 0 leads to k(L_{n}) ≃ k(A_{n−1}^{⊗2} ⊕ U_{n})(t_{1}, …, t_{n}) as S_{n}-fields. Therefore,

k(L_{n}) ≃ k(A_{n−1}^{⊗2} ⊕ U_{n})(t_{1}, …, t_{n})
= k(A_{n−1}^{⊗2} ⊕ U_{n} ⊕ ℤ)(t_{1}, …, t_{n−1})
≃ k(⋀^{2} A_{n−1})(x_{1}, …, x_{m}, t_{1}, …, t_{n−1})

as S_{n}-fields, which implies that

Z(n) ≃ k(L_{n})^{S^{n}} ≃ k(⋀^{2} A_{n−1})^{S^{n}}(x_{1}, …, x_{m}, t_{1}, …, t_{n−1}) ;

so Z(n) is rational over F_{0} = k(⋀^{2} A_{n−1})^{S^{n}}. □

5. Proof of Theorem 1.2

5.1. Proof of part (a). Reduction 5.1. Suppose an algebra A_{0} of degree n has the rational specialization property (see Section 2.4). If Theorem 1.2(a) holds for A_{0} then it holds for any central simple algebra A of degree n.

Proof. Suppose the for some r ≥ 1, A_{0}(t_{1}, …, t_{r}) is defined over a rational extension F_{0} of k. Let A/F be an arbitrary central simple algebra of degree n. Then by the rational specialization property, A_{0} embeds in A(t_{r+1}, …, t_{s}) for some s ≫ 0; thus A_{0}(t_{1}, …, t_{r}) embeds in A(t_{1}, …, t_{s}). This shows that A(t_{1}, …, t_{s}) is defined over F_{0}, as desired. □

In particular, in proving Theorem 1.2(a), we may assume that A is a division algebra of degree n; see Example 2.6. By primary decomposition (cf., e.g., [Pi, p. 261]), we only need to consider the cases where n = 2, n = 4 and n is an odd prime. This follows from the next reduction: Reduction 5.2. If the conclusion of Theorem 1.2(a) holds for central simple algebras A_{1}/F and A_{2}/F (for every choice of the base field k ⊂ F) then it also holds for A = A_{1} ⊗_{F} A_{2}.

Proof. After replacing A_{1} and A_{2} by, respectively, A_{1}(t_{1}, …, t_{s}) and A_{2}(t_{1}, …, t_{s}), we may assume that A_{1} is defined over a subfield F_{1} ⊂ F such that k ⊂ F_{1} and F_{1} is rational over k. We will now think of F_{1} (rather than k) as our new base field. After adding more indeterminates, we may assume that A_{2} is defined over a subfield F_{2} ⊂ F, where F_{1} ⊂ F_{2} and F_{2} is rational over F_{1}. Now F_{2} is rational over k, and since A_{1} and A_{2} are both defined over F_{2}, so is A. □

We are now ready to complete the proof of Theorem 1.2(a).

First, suppose n = 2 or 4. Since UD(n) has the rational specialization property, we may assume A = UD(n); see Reduction 5.1. But since the center of UD(n) is known to be rational for n = 2 (see [Pr, Theorem 2.2]) and n = 4 (see [F_{2}]), these algebras clearly satisfy the conclusion of Theorem 1.2(a). This completes the proof of the theorem for n = 2 and 4. We remark that the same argument goes through for n = 3 (because the center of UD(3) is known to be rational; see [F_{1}]) and for n = 5, 7 (because the centers of UD(5) and UD(7) are known to be stably rational; see [BL]), but we shall not need it in these cases.

From now on we will assume that n = p is an odd prime. Then the S_{n}-lattice A_{n−1}^{⊗2} is faithful; see Lemma 2.2. Furthermore, by a theorem of Bessenrodt and LeBruyn [BL, Proposition 3] (see also [Be, Lemma 2.8] for a more explicit form of this result), A_{n−1}^{⊗2} is permutation projective, i.e., there exists an S_{n}-lattice L such that P = A_{n−1}^{⊗2} ⊕ L is permutation. We can assume that k(P)^{S^{n}} is rational over k. Indeed, after adding a copy of U_{n} if necessary, we have P = U_{n} ⊕ Q for some permutation lattice Q, and so k(P) ≃ k(U_{n})(Q). Proposition 2.4 implies that k(P)^{S^{n}} is rational over k(U_{n})^{S^{n}}, which in turn is rational over k.

Let

i: A_{n−1}^{⊗2} ↪ k(A_{n−1}^{⊗2})^{∗}

and

j: A_{n−1}^{⊗2} ↪ k(A_{n−1}^{⊗2} ⊕ P)^{∗}

be the natural embeddings of S_{n}-modules. (Here, j identifies A_{n−1}^{⊗2} with the first summand of A_{n−1}^{⊗2} ⊕ P.) Recall that by Lemma 3.2 these embeddings, in combination with the exact sequence (2.4) (for G = S_{n} and H = S_{n−1}; see (4.1)), give rise to central simple algebras Alg(i) and Alg(j).

By Theorem 3.5, Alg(i) has the rational specialization property in the class of S_{n}/S_{n−1}-crossed products. Thus, the universal division algebra UD(n), being an S_{n}/S_{n−1}-crossed product (see Example 3.1), is a rational specialization of Alg(i). Since UD(n) has the rational specialization property in the class of all central simple algebras of degree n (see Example 2.6), so does Alg(i).

We claim that Alg(j) also has the rational specialization property in the class of central simple algebras of degree n. Indeed, by Lemma 3.2,

Alg(j) = Alg(i) ⊗_{F^{^{S}n}} E^{S^{n}} ,

where F = k(A_{n−1}^{⊗2}) and E = k(A_{n−1}^{⊗2} ⊕ P). Now Proposition 2.4 tells us that E^{S^{n}} is a rational extension of F^{S^{n}}, and the claim follows.

By Reduction 5.1 it now suffices to prove that Alg(j) is defined over a purely transcendental extension of k. Put E_{0} = k(A_{n−1}^{⊗2} ⊕ (0) ⊕ L) ⊆ E. Since the image of j is contained in E_{0}^{∗}, Lemma 3.3 tells us that Alg(j) is defined over E_{0}^{S_{n}}. But E_{0} ≃ k(P) and so E_{0}^{S_{n}} ≃ k(P)^{S^{n}} which is indeed rational over k. This completes the proof of Theorem 1.2(a). □

5.2. Proof of part(b). By the Merkurjev-Suslin Theorem,

M_{r}(A) = (a_{1}, b_{1})_{n_{1}} ⊗_{F} ⋯ ⊗_{F} (a_{l}, b_{l})_{n_{l}} ,

for some r, l ≥ 1, where (a, b)_{n} denotes a symbol algebra; see (1.4).

Let λ_{1}, …, λ_{l}, μ_{1}, …, μ_{l} be 2l central variables, algebraically independent over F. We will write λ in place of (λ_{1}, ..., λ_{l}) and μ in place of (μ_{1}, ..., μ_{l}). Then

M_{r}(A)(λ, μ) = (a_{1}, b_{1})_{n_{1}} ⊗_{K(λ,μ)} ⋯ ⊗_{K(λ,μ)} (a_{l}, b_{l})_{n_{l}} =
(a′_{1}, b′_{1})_{n_{1}} ⊗_{K(λ,μ)} ⋯ ⊗_{K(λ,μ)} (a′_{l}, b′_{l})_{n_{l}} =
((a′_{1}, b′_{1})_{n_{1}} ⊗_{F_{0}} ⋯ ⊗_{F_{0}} (a′_{l}, b′_{l})_{n_{l}}) ⊗_{F_{0}} K(λ, μ) ,

where a′_{i} = a_{i}λ_{i}^{n_{i}} and b′_{i} = b_{i}μ_{i}^{n_{i}} for i = 1, …, l and F_{0} = k(a′_{1}, b′_{1}, ..., a′_{l}, b′_{l}). This shows that M_{r}(D)(λ, μ) is defined over F_{0}. It remains to prove that F_{0} is rational over k. The 2l elements a′_{1}, b′_{1}, …, a′_{l}, b′_{l} are clearly algebraically independent over F. Hence, they are algebraically independent over k, and consequently, F_{0} is rational over k, as claimed. □

Remark 5.3. Our proof of Theorem 1.2 can be used to deduce explicit lower bounds on s in parts (a) and (b) from explicit lower bounds in Theorems of Bessenrodt-LeBruyn [BL, Proposition 3] (on rank(L)) and Merkurjev-Suslin (on r). The lowest possible value of r in part (b), called the Merkurjev-Suslin number, is of independent interest; see [Row_{3}, Section 7.2].

6. Proof of Theorem 1.3

Reduction 6.1. In the course of proving Theorem 1.3, we may assume without loss of generality that A is a division algebra.

Indeed, let D = Alg(μ), as in Theorem 3.5, with G = ℤ/m×ℤ/2, and H = {1}. Then D is a division algebra (see Remark 3.6), and any other G-crossed product A/F is a rational specialization of D. Thus, if we know that Theorem 1.3 holds for D then it holds for A(t_{1}, …, t_{s}), where t_{1}, …, t_{s} are independent variables over F. Using induction on s, we see that Reduction 6.1 is now a consequence of the following lemma (applied to B = M_{m}(A), with r = 2, m_{1} = m and m_{2} = 2m): Lemma 6.2. Let B/K be a central simple algebra of degree d = m_{1} … m_{r} and let t be an independent variable over K. Assume K contains a primitive root of unity of degree lcm(m_{1}, …, m_{r}). If

B(t) = (a_{1}(t), b_{1}(t))_{m_{1}} ⊗ ⋯ ⊗ (a_{r}(t), b_{r}(t))_{m_{r}}

for some a_{1}(t), b_{1}(t), …, a_{r}(t), b_{r}(t) ∈ K(t) then

B = (a′_{1}, b′_{1})_{m_{1}} ⊗ ⋯ ⊗ (a′_{r}, b′_{r})_{m_{r}}

for some a′_{1}, b′_{1}, …, a′_{r}, b′_{r} ∈ K.

Our proof is based on a standard specialization argument; for the sake of completeness, we supply the details below.

Proof. We may assume that K is an infinite field; otherwise B is a matrix algebra over K, and we can take, e.g., a′_{i} = 1, b′_{i} = −1 for every i.

Choose generators x_{i}(t) and y_{i}(t) for the cyclic subalgebra (a_{i}(t), b_{i}(t))_{m_{i}} of B(t) = B ⊗_{K} K(t) such that x_{i}(t)^{m_{i}} = a_{i}(t), y_{i}(t)^{m_{i}} = b_{i}(t), and x_{i}(t)y_{i}(t) = ζ_{m_{i}}y_{i}(t)x_{i}(t), where ζ_{m_{i}} is a primitive root of unity of degree m_{i} in K. Choose a K-basis b_{1}, …, b_{d^{2}} of B and write

x_{i}(t) = ∑_{j=1}^{d^{2}} α_{ij}(t)b_{i} and y_{i}(t) = ∑_{j=1}^{d^{2}} β_{ij}(t)b_{i} ,  (6.1)

for some α_{ij}(t), β_{ij}(t) ∈ K(t). Since K is an infinite field, we can choose t_{0} ∈ K such that α_{ij}(t_{0}) and β_{ij}(t_{0}) are well-defined and

x_{i}(t_{0}) = ∑_{j=1}^{d^{2}} α_{ij}(t_{0})b_{i} and y_{i}(t_{0}) = ∑_{j=1}^{d^{2}} β_{ij}(t_{0})b_{i}

are non-zero. Let B_{i} denote the subalgebra of B that is generated by x_{i}(t_{0}) and y_{i}(t_{0}). Then B_{i} = (a′_{i}, b′_{i})_{m_{i}}, where a′_{i} = x_{i}(t_{0})^{m_{i}} = a_{i}(t_{0}) and b′_{i} = y_{i}(t_{0})^{m_{i}} = b_{i}(t_{0}), and B_{1}, …, B_{r} are commuting subalgebras of B of degrees m_{1}, …, m_{r}. Hence, by the double centralizer theorem (cf., e.g., [Pi, Theorem 12.7]), B = B_{1}⊗ ⋯ ⊗ B_{r}. This completes the proof of Lemma 6.2 and thus of Reduction 6.1. □

We now continue with the proof of Theorem 1.3. In the course of the proof we shall use the following notations. Write G = ℤ/m × ℤ/2 = ⟨σ_{1}, σ_{2}⟩, where σ_{1}^{m} = σ_{2}^{2} = 1. Let K = F (α_{1}, α_{2}), be a maximal G-Galois subfield of A, where α_{1}^{m} = a_{1} and α_{2}^{2} = a_{2} are elements of F, and

σ_{1}(α_{1}) = ζ_{m}α_{1} ,  σ_{1}(α_{2}) = α_{2} ,
σ_{2}(α_{1}) = α_{1} ,  σ_{2}(α_{2}) = −α_{2} .  (6.2)

Here ζ_{m} ∈ F is a primitive m^{th} root of unity, so that K is, indeed, a G-Galois extension of F. Note that the statement of Theorem 1.3 assumes that F contains not only a primitive mth root of unity ζ_{m} but also a primitive 2m^{th} root of unity ζ_{2m}; we shall make use of ζ_{2m} later in the proof.

By the Skolem-Noether Theorem, there exist units z_{1}, z_{2} ∈ A such that z_{i}xz_{i}^{−1} = σ_{i}(x) for every x ∈ K (i = 1, 2). Set

z_{1}^{m} = b_{1} ∈ F (α_{2})^{∗}, z_{2}^{2} = b_{2} ∈ F (α_{1})^{∗}, and u = z_{1}z_{2}z_{1}^{−1}z_{2}^{−1} ∈ K^{∗} .  (6.3)

By [AS, Theorem 1.3], the algebra structure of A can be recovered from the G-field K and the elements u ∈ K^{∗}, b_{1} ∈ F (α_{2})^{∗} and b_{2} ∈ F (α_{1})^{∗}. (These elements have to satisfy certain compatibility conditions; the exact form of these conditions shall not concern us in the sequel.) We will write A = (K, G, u, b_{1}, b_{2}). Lemma 6.3. Let A = (K, G, u, b_{1}, b_{2}) and A′ = (K, G, u′, b′_{1}, b′_{2}) be G-crossed products. Then A ⊗_{F} A′ is Brauer equivalent to (K, G, uu′, b_{1}b′_{1}, b_{2}b′_{2}).

Proof. The class of A = (K, G, u, b_{1}, b_{2}) in the relative Brauer group B(K/F) = H^{2}(G, K^{∗}) is given by a normalized 2-cocycle a: G × G → K^{∗} so that

b_{i} = a(σ_{i}, σ_{i})a(σ_{i}^{2}, σ_{i}) … a(σ_{i}^{m_{i}−1}, σ_{i})

holds for i = 1, 2, where m_{1} = m and m_{2} = 2, and

u = a(σ_{1}, σ_{2})a(σ_{s}, σ_{1})^{−1} .

Similarly, the class of A′ is given by a 2-cocycle a′. Then the class of A ⊗_{F} A′ is given by the cocycle aa′; see, e.g., [Pi, Proposition 14.3]. This proves the lemma.

The following alternative ring-theoretic argument was suggested by the referee: Choose z_{1}, z_{2} ∈ A, as in (6.3), and similarly for z′_{1}, z′_{2} in A′. The subalgebra S of A ⊗_{F} A′ generated by K ⊗ 1, z_{1} ⊗ z′_{1} and z_{2} ⊗ z′_{2}, is clearly isomorphic to (K, G, uu′, b_{1}b′_{1}, b_{2}b′_{2}). Its centralizer C_{A⊗A′}(S) is an F -central simple algebra of degree 2m = [K : F], containing (K ⊗ K)^{G}, where G acts diagonally on K ⊗ K. Since G is abelian, (K ⊗ K)^{G} ≃ F ⊕ ⋯ ⊕ F, as an F [G]-algebra. In particular, C_{A⊗A′} contains the idempotents of K ⊗ K and, hence, is split over F. We thus conclude that

A ⊗_{F} A′ ≃ S ⊗_{F} C_{A⊗_{F} A′}(S) ∼ S ≃ (K, G, uu′, b_{1}b′_{1}, b_{2}b′_{2}) ,

as claimed. (Here ∼ denotes Brauer equivalence over F.) □

We now proceed with the proof of Theorem 1.3, using the notations of (6.2) and (6.3). Since b_{1} = z_{1}^{m} ∈ K^{σ_{1}} = F (α_{2}), we can write

b_{1} = f_{1} + f_{2}α_{2} ,  (6.4)

for some f_{1}, f_{2} ∈ F. Lemma 6.4. (a) If f_{1} = 0 then A is cyclic.

(b) If f_{2} = 0 then A = (a, b)_{m} ⊗ (c, d)_{2}, for some a, b, c, d ∈ F^{∗}.

Proof. (a) If f_{1} = 0 then z_{1}^{2m} = b_{1}^{2} = f_{2}^{2}a_{2} ∈ F^{∗} but z_{1}^{m} = f_{2}α_{2} ∉ F. Since F contains a primitive root of unity of degree 2m, F (z_{1}) is a cyclic maximal subfield of A of degree 2m; cf. [Lang, Theorem VIII.6.10(b)]. Thus A is a cyclic algebra, as claimed.

(b) If f_{2} = 0, i.e., b_{1} ∈ F, then the F -subalgebra A_{0} of A generated by z_{1} and α_{1} is cyclic of degree m. By the double centralizer theorem, A = A_{m} ⊗ Q, where Q is a quaternion algebra, as claimed. □

We are now ready to finish the proof of Theorem 1.3. Lemma 6.4 tells us that Theorem 1.3 is immediate if f_{1} = 0 or f_{2} = 0. Thus from now on we shall assume f_{1}f_{2} ≠ 0.

Now let A = (K, G, u, b_{1}, b_{2}) and, for any f ∈ F^{∗}, define A_{f} = (K, G, u, fb_{1}, b_{2}). Since (a_{1}, f)_{m}⊗_{F} M_{2}(F) ≃ (K, G, 1, f, 1), Lemma 6.3 tells us that A_{f} ∼ (a_{1}, f)_{m}⊗_{F} A, where ∼ denotes Brauer equivalence. In other words, A ∼ (f, a_{1})_{m} ⊗_{F} A_{f}. Thus it is enough to show that A_{f} is cyclic, for some f ∈ F^{∗}.

To prove the last assertion, observe that if we expand (z_{1} + α_{1})^{m} then all terms, other than z_{1}^{m} and α_{1}^{m}, will cancel. (For a simple proof of this fact, due to Bergman, see [Row_{3}, p. 195]). Thus, if γ = z_{1} + α_{1} then

γ^{m} = z_{1}^{m} + α_{1}^{m} = fb_{1} + a_{1}

in A_{f}. Setting f = −\frac{a_{1}}{f_{1}} ∈ F^{∗}, we obtain γ^{m} = cα_{2}, where c = −\frac{a_{1}f_{2}}{f_{1}} ∈ F^{∗}; see (6.4). Thus γ^{2m} = c^{2}a_{2} ∈ F^{∗} but γ^{m} ∉ F. Since F contains a primitive 2m^{th} root of unity, F (γ)/F is a cyclic field extension of degree 2m. In other words, F (γ) a cyclic maximal subfield of A_{f}, and A_{f} is a cyclic algebra of degree 2m, as claimed. □

7. The field of definition of a quadratic form

7.1. Preliminaries. Let V = F^{n} be an F -vector space, equipped with a quadratic form q: V → F. Recall that q is said to be defined over a subfield F_{0} of F if q = q_{F_{0}} ⊗ F, where q_{F_{0}} is a quadratic form on V_{0} = F_{0}^{n}. Is easy to see that q is defined over F_{0} if and only if V has an F -basis e_{1}, …, e_{n} such that b(e_{i}, e_{j}) ∈ F_{0}, where b: V × V → F is the symmetric bilinear form associated to q (i.e., q(v) = b(v, v)).

We shall always assume that char(F) ≠ 2 and F (and F_{0}) contain a base subfield k. As usual, we shall write <a_{1}, …, a_{n}> for the diagonal form

a_{n}x_{1}^{2} + ⋯ + a_{n}x_{n}^{2}

and ≪ a_{1}, …, a_{n} ≫ for the Pfister form <1, a_{1}> ⊗ ⋯ ⊗ <1, a_{n}>. Given a quadratic form q we shall ask:

(a) What is the smallest value of trdeg_{k}(F_{0}), where q is defined over F_{0}? We shall denote this number by τ(q).

(b) Can q be defined over a rational extension F_{0} of k? These are the same questions we asked for central simple algebras in the Introduction. In the case of quadratic forms our answers are more complete (and the proofs are easier). Proposition 7.1. Let V = F^{n} and let q: V → F be a quadratic form on V. Then

(a) τ(q) ≤ n. Moreover, if a_{1}, …, a_{n} are independent variables over k, F = k(a_{1}, …, a_{n}), and q = <a_{1}, …, a_{n}> then τ(q) = n.

(b) Let t_{1}, …, t_{n} be independent variables over F. Then q′ = q⊗_{F} F (t_{1}, …, t_{n}) is defined over a rational extension F_{0} of k.

Proof. Diagonalizing q, write q = <a_{1}, …, a_{n}> in the basis e_{1}, …, e_{n}.

(a) To prove the first assertion, set F_{0} = k(a_{1}, …, a_{n}). Then q is defined over F_{0} = k(a_{1}, …, a_{n}) and trdeg_{k}(F_{0}) ≤ n, as desired. For the proof of the second assertion see [Re_{2}, Proof of Theorem 10.3].

(b) Set a′_{i} = t_{i}^{2}a_{i}. Then q′ = <a_{1}, …, a_{n}> = <a′_{1}, …, a′_{n}> over F (t_{1}, …, t_{n}). Hence, q′ is defined over F_{0} = k(a′_{1}, …, a′_{n}). We claim that F_{0} is rational over k. Indeed, since the nonzero elements of {a′_{1}, …, a′_{n}} are algebraically independent over F, they are algebraically independent over k, and the claim follows. □

In the sequel we shall need the following analogue of Lemma 2.7. Lemma 7.2. Let q be a quadratic form defined over F, t_{1}, …, t_{r} be independent variables over F, and F′ = F (t_{1}, …, t_{r}). Set q′ = q ⊗_{F} F′. Then τ(q) = τ(q′).

Proof. The inequality τ(q′) ≤ τ(q) is obvious from the definition of τ(q). To prove the opposite inequality, we may assume F is an infinite field; otherwise τ(q) = 0, and there is nothing to prove. We may also assume r = 1; the general case will then follows by induction on r. Let b′ be the symmetric bilinear form associated to q′ and choose a basis b_{1}(t), …, b_{n}(t) of (F′)^{n} so that trdeg_{k} k(α_{ij}(t)) = τ(q′), where α_{ij}(t) = b′(b_{i}(t), b_{j}(t)). Since F is an infinite field, we can find a c ∈ F such that (i) the vectors b_{1}(c), …, b_{d}(c) are welldefined and form a basis of F^{d}, and (ii) each α_{ij}(c) is well-defined. Now q is defined over k(α_{ij}(c)) and thus

τ(q) ≤ trdeg_{k}(α_{ij}(c)) ≤ trdeg_{k}(α_{ij}(t)) = τ(q′) ,

as claimed. □

7.2. Proof of Theorem 1.5. Let F be a field containing a primitive 4^{th} root of unity. Note that for the purpose of proving Theorem 1.5, we may assume that A/F is a division algebra. Otherwise, A is isomorphic to M_{4}(F) or to M_{2}(D), where D = (a, b)_{2} is a quaternion algebra. Thus, A is defined over k or over the field F_{0} = k(a, b), respectively, and so is the trace form of A. Alternatively, a simple direct computation shows that the trace form of M_{2}(E) is trivial (and thus is defined over k) for any central simple algebra E/F.

From now on we will assume that A/F is a division algebra of degree 4. By a theorem of Albert [A], A is a G-crossed product, with G = ℤ/2 × ℤ/2. Let K be a G-Galois maximal subfield. Using the notations introduced in Section 6 (with m = 2), we write G = <σ_{1}, σ_{2}>, K = F (α_{1}, α_{2}), α_{i}^{2} = a_{i} ∈ F and A = (K, G, u, b_{1}, b_{2}) for some u ∈ K^{∗}, b_{1} ∈ F (α_{2}) = K^{σ_{1}}, and b_{2} ∈ F (α_{1}) = K^{σ_{2}}. Set σ_{3} = σ_{1}σ_{2} ∈ Gal(K/F); z_{3} = (z_{1}z_{2})^{−1}, α_{3} = α_{1}α_{2}, a_{3} = α_{3}^{2} = a_{1}a_{2}, b_{3} = z_{3}^{2} (so that b_{i} = z_{i}^{2} for i = 1, 2, 3), and

t_{i} = \frac{1}{2} Tr_{K^{σ_{i}}/F} (z_{i}^{2}) , n_{i} = N_{K^{σ_{i}}/F} (z_{i}^{2})

for i = 1, 2, 3.

Our proof of Theorem 1.5 is based on the following result of Serre [Se], [RST]. Proposition 7.3. Suppose z_{1} and z_{2} are chosen so that t_{i} ≠ 0 and n_{i}^{2} − t_{i} ≠ 0 for any i = 1, 2, 3. Then the trace form q of A is Witt-equivalent (over F) to q_{2} ⊕ q_{4}, where

q_{2} =≪ n_{1} − t_{1}^{2}, n_{2} ≫

is a 2-fold Pfister form and

q_{4} =≪ t_{1} − n_{1}^{2}, (n_{2} − t_{2}^{2})n_{2}, t_{1}t_{2}, t_{2}t_{3} ≫

is a 4-fold Pfister form. □

We claim that for the purpose of proving Theorem 1.5, we may assume without loss of generality that t_{i} ≠ 0 and n_{i}^{2} − t_{i} ≠ 0 for any i = 1, 2, 3. Indeed, in view of Lemma 7.2 it suffices to prove Theorem 1.5 for a single division algebra A which has the rational specialization property in the class of algebras of degree 4, e.g., for A = UD(4); see Remark 2.8. Thus we only need to show that in this algebra t_{i} ≠ 0 and n_{i} − t_{i}^{2} ≠ 0 for any choice of z_{1}, z_{2}.

Indeed, we may assume without loss of generality that i = 1 (the cases where i = 2 and 3 will then follow by symmetry). Write b_{1} = f_{1} + f_{2}α_{2} for some f_{1}, f_{2} ∈ F, where t_{1} = f_{1} and n_{1} − t_{1}^{2} = f_{2}^{2}a_{2}. Lemma 6.4 shows that if t_{1} = 0 then A is cyclic and if n_{1} − t_{1}^{2} = 0 then A is biquaternion. But since our algebra A has the rational specialization property, it is neither cyclic nor biquaternion. We conclude that t_{1}(n_{1} − t_{1}^{2}) ≠ 0, as claimed.

We now proceed to simplify the form given by Proposition 7.3. After expanding q_{2} and q_{4}, cancelling the common term <1, t_{1}^{2} − n_{1}> (which can be done, since we are assuming \sqrt{−1} ∈ F) and dividing some of the entries by elements of (F^{∗})^{2}, we see that the trace form of A is Witt equivalent to the 16-dimensional form

q = <1 , 1 − \frac{n_{1}}{t_{1}^{2}} > ⊗ (<\frac{n_{2}}{t_{2}^{2}} >⊕ ≪ (1 − \frac{n_{2}}{t_{2}^{2}} )\frac{n_{2}}{t_{2}^{2}} , t_{1}t_{2}, t_{2}t_{3} ≫_{0}) ,  (7.1)

where ≪ λ_{1}, …, λ_{r} ≫_{0} is defined as a 2^{r} − 1-dimensional form such that ≪ λ_{1}, λ_{2}, …, λ_{r} ≫_{0} ⊕<1> is the n-fold Pfister form ≪ λ_{1}, λ_{2}, …, λ_{r} ≫.

Note that since q and the trace form of A are Witt equivalent 16-dimensional forms, the Witt Decomposition Theorem implies that they are, in fact, the same (i.e., isometric). We now observe that all entries of q lie in the subfield F_{0} = k(\frac{n_{1}}{t_{1}^{2}} , \frac{n_{2}}{t_{2}^{2}} , t_{1}t_{2}, t_{2}t_{3}) of F. Thus the trace form of A is defined over F_{0}. Since F_{0} is generated by 4 elements over k, we have trdeg_{k} F_{0} ≤ 4. This completes the proof of Theorem 1.5. □

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Department of Mathematics, Temple University, Philadelphia, PA 19122-6094

E-mail address: lorenz@math.temple.edu
URL: http://www.math.temple.edu/^{∼}lorenz

Department of Mathematics, University of British Columbia, Vancouver, BC, Canada, V6T 1Z2

E-mail address: reichst@math.ubc.ca
URL: http://www.math.ubc.ca/^{∼}reichst

Department of Mathematics and Computer Science, Bar-Ilan University, Ramat-Gan 52900, Israel

E-mail address: rowen@macs.biu.ac.il

Department of Mathematics, University of Texas, Austin, TX 78712

E-mail address: saltman@math.utexas.edu