Cohomological Invariants for G-Galois Algebras

and Self-Dual Normal Bases

E. Bayer-Fluckiger and R. Parimala

Received: August 26, 2016

Communicated by Nikita Karpenko

Abstract. We define degree two cohomological invariants for G-Galois algebras over fields of characteristic not 2, and use them to give necessary conditions for the existence of a self–dual normal basis. In some cases (for instance, when the field has cohomological dimension ≤ 2) we show that these conditions are also sufficient.

Introduction

Let k be a field of characteristic ≠ 2, and let L be a finite degree Galois extension of k. Let G = Gal(L/k). The trace form of L/k is by definition the quadratic form q_{L} : L × L → k defined by q_{L}(x, y) = Tr_{L/k}(xy). Note that q_{L} is a G-quadratic form, in other words we have q_{L}(gx, gy) = q_{L}(x, y) for all x, y ∈ L. A normal basis (gx)_{g∈G} of L over k is said to be self-dual if q_{L}(gx, gx) = 1 and q_{L}(gx, hx) = 0 if g ≠ h. It is natural to ask which extensions have a self-dual normal basis. This question is investigated in several papers (see for instance [BL 90], [BSe 94], [BPS 13]). It is necessary to work in a more general context than the one of Galois extensions, namely that of G-Galois algebras (see for instance [BSe 94], §1); one advantage being that this category is stable by base change of the ground field; the notion of a self-dual normal basis is defined in the same way.

If k is a global field, then the Hasse principle holds : a G-Galois algebra has a self-dual normal basis over k if and only if such a basis exists everywhere locally (see [BPS 13]). The present paper completes this result by giving necessary and sufficient conditions for the existence of a self-dual normal basis when k is a local field (cf. §7). The conditions are given in terms of cohomological invariants defined over the ground field k constructed in §3 and §4.

For an arbitrary ground field k, we start with the H^{1}-invariants defined in [BSe 94], §2. Recall from [BSe 94] that the vanishing of these invariants is a necessary condition for the existence of a self-dual normal basis; it is also sufficient in the case of fields of cohomological dimension 1 (see [BSe 94], Corollary 2.2.2 and Proposition 2.2.4).

Let k[G] be the group algebra of G over k, and let J be its radical; the quotient k[G]^{s} = k[G]/J is a semisimple k-algebra. Let σ : k[G] → k[G] be the k-linear involution sending g to g^{−1}; it induces an involution σ^{s} : k[G]^{s} → k[G]^{s}. The algebra k[G]^{s} splits as a product of simple algebras. If A is a σ^{s}-stable simple algebra which is a factor of k[G]^{s}, we denote by σ_{A} the restriction of σ^{s} to A, and by E_{A} the subfield of the center of A fixed by σ_{A}. We say that A is orthogonal if σ_{A} is the identity on the center of A, and if over a separable closure of k it is induced by a symmetric form, and unitary if σ_{A} is not the identity on the center of A (see 1.3 for details).

Let L be a G-Galois algebra over k, and let us assume that its H^{1}-invariants are trivial. We then define, for every orthogonal or unitary A as above, cohomology classes in H^{2}(k, Z/2Z), denoted by c_{A}(L) in the orthogonal case and by d_{A}(L) in the unitary case (see §3 and §4). They are invariants of the G-Galois algebra L. They also provide necessary conditions for the existence of a self-dual normal basis (this involves restriction to certain finite degree extensions of k, namely, the extensions E_{A}/k; see Propositions 3.5 and 4.7 for precise statements). If moreover k has cohomological dimension ≤ 2, then these conditions are also sufficient (Theorem 5.3.). Finally, if k is a local field, then the conditions can be expressed in terms of the invariants c_{A}(L) and d_{A}(L), without passing to finite degree extensions (Theorem 7.1). Section 8 applies the results of §7 and the Hasse principle of [BSP 13] to give necessary and sufficient conditions for the existence of a self-dual normal basis when k is a global field (Theorem 8.1).

Section 6 deals with the case of cyclic groups of order a power of 2 over arbitrary fields. We show that at most one of the unitary components A gives rise to a non-trivial invariant d_{A}(L) (Proposition 6.4 (i)), and that this invariant provides a necessary and sufficient condition for the existence of a self-dual normal basis (Corollary 6.5).

Acknowledgment : The first named author is partially supported by grant 200021-163188 of the Swiss National Science Foundation, and the second named author is partially supported by National Science Foundation grant DMS-1401319.

§1. Definitions, notation and basic facts

1.1. Galois cohomology

We use standard notation in Galois cohomology. If K is a field, we denote by K_{s} a separable closure of K, and by Γ_{K} the Galois group Gal(K_{s}/K). For any discrete Γ_{K}-module C, set H^{i}(K, C) = H^{i}(Γ_{K}, C). If Γ is a finite or profinite group, set H^{i}(Γ) = H^{i}(Γ, Z/2Z). If U is a K-group scheme, we denote by H^{1}(K, U) the pointed set H^{1}(Γ_{K}, U(K_{s})).

1.2. Algebras with involution and unitary groups

Let K be a field of characteristic ≠ 2, and let R be a finite dimensional algebra over K. An involution of R is a K-linear anti-automorphism σ : R → R such that σ^{2} is the identity.

Let us denote by Comm_{K} the category of commutative K-algebras, and by Group the category of groups. If (R, σ) is an algebra with involution, the functor Comm_{K} → Group given by S ↦ {x ∈ R ⊗_{K} S | xσ(x) = 1} is the functor of points of a scheme over Spec(K); we denote it by U_{R,K}.

Let h = ⟨1⟩ be the rank one unit hermitian form over (R, σ), given by h(x, y) = xσ(y) for all x, y ∈ R. Then U_{R,K} is the scheme of automorphisms of the hermitian form h. This is a smooth, finitely presented affine group scheme over Spec(K) (see for instance [BF 15], Appendix A). Moreover, H^{1}(K, U_{R,K}) is in natural bijection with the set of isomorphism classes of rank one hermitian forms over (R, σ) that become isomorphic to h over K_{s} (see [Se 64], chap. III, §1).

If F is a subfield of K, then U_{R,F} = R_{K/F} (U_{R,K}), where R_{K/F} denotes Weil restriction of scalars relative to the extension K/F.

Let Z be the center of R, and assume that R is a simple algebra. We say that (R, σ) is a central simple algebra with involution over K if the fixed field of σ in Z is equal to K. If (R, σ) is central simple algebra with involution over K, we set U_{R} = U_{R,K}.

1.3. Dévissage

Let G be a finite group and let k[G] be its group algebra over k. The canonical involution of k[G] is the k-linear involution σ : k[G] → k[G] such that σ(g) = g^{−1} for all g ∈ G. Let J be the radical of k[G], and set k[G]^{s} = k[G]/J; it is a semisimple k-algebra. Since J is stable by σ, we obtain an involution σ^{s} : k[G]^{s} → k[G]^{s}. Set U_{G} = U_{k[G],k} and U_{G}^{r} = U_{k[G]^{s},k}. Let N be the kernel of the natural surjection U_{G} → U_{G}^{r}. Let us define group schemes N_{i} by setting N_{i}(S) = {x ∈ N(S) | x ≡ 1 mod J^{i} ⊗_{k} S}. Then 1 = N_{m} ⊂ N_{m−1} ⊂ ⋯ ⊂ N_{1} = N, where m is an integer such that J^{m} = 0. Note that J^{i}/J^{i+1} is a module over the semisimple algebra k[G]^{s}, hence N_{i}/N_{i−1} is isomorphic to a finite product of additive groups G_{a}; therefore N is a split unipotent group. This implies that H^{1}(k, U_{G}) = H^{1}(k, U_{G}^{r}) (see for instance [Sa 81], Lemme 1.13).

The semisimple algebra k[G]^{s} is known to be a direct product of simple algebras. Note that k[G] comes by scalar extension from k_{0}[G] for k_{0} = Q or F_{p}, hence the centers of the factors of k[G]^{s} are abelian Galois extensions of k of finite degree; some are stable under σ^{s} (we call them A), and others come in pairs, interchanged by σ^{s} (we call them B).

If A is a σ^{s}-stable simple factor of k[G]^{s}, we denote by σ_{A} the restriction of σ^{s} to A, by F_{A} the center of A, and by E_{A} the subfield of σ_{A}-invariant elements of F_{A}. Note that U_{A} is a group scheme over Spec(E_{A}). Similarly, if B is the product of two simple algebras interchanged by σ^{s}, we denote by E_{B} the subfield of the center of B fixed by the involution; U_{B,E_{B}} is a group scheme over Spec(E_{B}).

We have U_{G}^{r} ≃ ∏_{A} R_{E_{A}/k}(U_{A}) × ∏_{B} R_{E_{B}/k}(U_{B,E_{B}}), hence

H^{1}(k, U_{G}^{r}) = ∏_{A} H^{1}(k, R_{E_{A}/k}(U_{A})) × ∏_{B} H^{1}(k, R_{E_{B}/k}(U_{B,E_{B}})).

Note that H^{1}(k, R_{E_{B}/k}(U_{B,E_{B}})) = H^{1}(E_{B}, U_{B,E_{B}}) = 0 (see for instance [KMRT 98], (29.2)), that H^{1}(k, R_{E_{A}/k}(U_{A})) = H^{1}(E_{A}, U_{A}) (see for instance [O 84], 2.3), and that H^{1}(k, U_{G}) = H^{1}(k, U_{G}^{r}) (see above). Therefore we have

H^{1}(k, U_{G}) = ∏_{A} H^{1}(E_{A}, U_{A}).

The algebras with involution (A, σ_{A}) appearing in this product are of three types :

(a) The involution σ_{A} : A → A is not the identity on the center F_{A} of A. Hence F_{A}/E_{A} is a quadratic extension. Such an algebra with involution is called unitary; the group scheme U_{A} is of Dynkin type A.

(b) The involution σ_{A} : A → A is the identity on F_{A} (which is then equal to E_{A}), and, over a separable closure of E_{A}, the involution is induced by a symmetric form. Such an algebra with involution is called orthogonal; the group scheme U_{A} is of Dynkin type B or D.

(c) The involution σ_{A} : A → A is the identity on F_{A} (which is then equal to E_{A}), and, over a separable closure of E_{A}, the involution is induced by a skew-symmetric form. Such an algebra with involution is called symplectic; the group scheme U_{A} is of Dynkin type C.

1.4. G-quadratic forms

A G-quadratic form is a pair (M, q), where M is a k[G]-module that is a finite dimensional k-vector space, and q : M ×M → k is a non-degenerate symmetric bilinear form such that

q(gx, gy) = q(x, y)

for all x, y ∈ M and all g ∈ G. We say that two G-quadratic forms (M, q) and (M′, q′) are isomorphic if there exists an isomorphism of k[G]-modules f : M → M′ such that q′(f(x), f(y)) = q(x, y) for all x, y ∈ M. If this is the case, we write (M, q) ≃_{G} (M′, q′), or q ≃_{G} q′. It is well-known that Gquadratic forms correspond bijectively to non-degenerate hermitian forms over (k[G], σ) (see for instance [BPS 13], 2.1, Example on page 441). The unit Gform is by definition the pair (k[G], q_{0}), where q_{0} is the G-form characterized by q(g, g) = 1 and q(g, h) = 0 if g ≠ h, for g, h ∈ G.

1.5. Trace forms of G-Galois algebras

If L is an étale k-algebra, we denote by

q_{L} : L × L → k, q_{L}(x, y) = Tr_{L/k}(xy),

its trace form. Then q_{L} is a non-degenerate quadratic form over k; if moreover L is a G-Galois algebra, then q_{L} is a G-quadratic form.

Let L be a G-Galois algebra; then L has a self-dual normal basis over k if and only if q_{L} is isomorphic to q_{0} as a G-quadratic form. Let ϕ : Γ_{k} → G be a continuous homomorphism corresponding to L (see for instance [BSe 94], 1.3). Recall that ϕ is unique up to conjugation. The composition

Γ_{k} →^{ϕ} G → U_{G}(k) → U_{G}(k_{s})

is a 1-cocycle Γ_{k} → U_{G}(k_{s}). Let u(L) be its class in the cohomology set H^{1}(k, U_{G}); it does not depend on the choice of ϕ. The G-Galois algebra L has a self-dual normal basis over k if and only if u(L) = 0, cf. [BSe 94], Corollaire 1.5.2.

Recall from 1.3 that we have

H^{1}(k, U_{G}) = ∏_{A} H^{1}(E_{A}, U_{A}).

Let u_{A}(L) be the image of u(L) in H^{1}(E_{A}, U_{A}); note that L has a self-dual normal basis if and only if u_{A}(L) = 0 for every A.

Let A be as above. Composing the injection G → U_{G}(k) with the natural map U_{G}(k) → U_{G}^{r}(k) → R_{E_{A}/k}(U_{A})(k) = U_{A}(E_{A}), we obtain a homomorphism G → U_{A}(E_{A}), denoted by i_{A}.

Let ϕ_{A} : Γ_{E_{A}} → Γ_{k} → G be the composition of ϕ : Γ_{k} → G with the inclusion of Γ_{E_{A}} in Γ_{k}. Composing ϕ_{A} with the map i_{A} : G → U_{A}(E_{A}) defined above we obtain a 1-cocycle Γ_{E_{A}} → U_{A}(k_{s}). The class of this 1-cocycle in H^{1}(E_{A}, U_{A}) is equal to u_{A}(L).

§2. The H^{1}-condition

Let L be a G-Galois algebra over k, and let ϕ : Γ_{k} → G be a homomorphism corresponding to L. Let n be an integer ≥ 1. Then ϕ induces a homomorphism

ϕ^{∗} : H^{n}(G) → H^{n}(k, Z/2Z).

Note that ϕ^{∗} is independent of the choice of ϕ in its conjugacy class (see [Se 68], chap. VII, proposition 3). For all x ∈ H^{n}(G), set x_{L} = ϕ^{∗}(x).

Proposition 2.1. If L has a self-dual normal basis over k, then for all x ∈ H^{1}(G) we have x_{L} = 0.

Proof. See [BSe 94], Corollaire 2.2.2.

If cd_{2}(Γ_{k}) ≤ 1, then L has a self-dual normal basis over k if and only if x_{L} = 0 for all x ∈ H^{1}(G), see [BSe 94], Proposition 2.2.4.

We say that the H^{1}-condition is satisfied if x_{L} = 0 for all x ∈ H^{1}(G). Let G^{2} be the subgroup of G generated by the squares of elements of G. Note that G/G^{2} is an elementary abelian 2-group, and that the H^{1}-condition means that the homomorphism Γ_{k} → G → G/G^{2} induced by ϕ is trivial, i.e. ϕ(Γ_{k}) ⊂ G^{2}.

§3. Orthogonal invariants

We keep the notation of the previous sections. In particular, G is a finite group, L is a G-Galois algebra, and ϕ : Γ_{k} → G is a homomorphism corresponding to L. Let us suppose that the H^{1}-condition is satisfied.

Let A be an orthogonal σ^{s}-stable central simple factor of k[G]^{s} (see 1.3), and recall that the center of A is denoted by E_{A}. Let us denote by ⟨A⟩ the subgroup of Br(E_{A}) generated by the class of the algebra A. Note that since σ_{A} : A → A is an orthogonal involution, this class has order at most 2, hence ⟨A⟩ is a subgroup of Br_{2}(E_{A}).

The aim of this section is to define two invariants : an invariant c_{A}(L) ∈ H^{2}(k) of the G–Galois algebra L, and an invariant clif_{A}(q_{L}) ∈ Br_{2}(E_{A})/⟨A⟩ of the G-form q_{L}. We shall compare these two invariants (cf. Theorem 3.3), and give a necessary condition for the existence of self-dual normal bases (Corollary 3.5).

Let U_{A}^{0} be the connected component of the identity in U_{A}. Let i_{A} : G → U_{A}(E_{A}) be the homomorphism defined in 1.5, and let π : U_{A}(E_{A}) → U_{A}(E_{A})/U_{A}^{0}(E_{A}) be the projection. Since U_{A}(E_{A})/U_{A}^{0}(E_{A}) is of order ≤ 2, we have π(i_{A}(G^{2})) = 0; i.e. i_{A}(G^{2}) ⊂ U_{A}^{0}(E_{A}).

Let Ũ_{A} be the Spin group of (A, σ); note that if dim_{k}(A) ≥ 3, then Ũ_{A} is the universal cover of U_{A}^{0}. Let s : Ũ_{A} → U_{A}^{0} be the covering map. We have an exact sequence of algebraic groups over E_{A}

1 → Z/2Z → Ũ_{A} →^{s} U_{A}^{0} → 1.

Let us consider the associated cohomology exact sequence

Ũ_{A}(E_{A}) →^{s} U_{A}^{0}(E_{A}) →^{δ} H^{1}(E_{A}).

Lemma 3.1. We have i_{A}(G^{2}) ⊂ s(Ũ_{A}(E_{A})).

Proof. In view of the above exact sequence, it suffices to prove that δ(i_{A}(G^{2})) = 0. In order to prove this, let us first assume that A is not split. Then we have U_{A}(E_{A}) = U_{A}^{0}(E_{A}) (cf. [K 69], Lemma 1 b, see also [B 94], cor. 2). Since H^{1}(E_{A}) is a 2-torsion group and since i_{A}(G^{2}) ⊂ U_{A}^{0}(E_{A}), this implies that δ(i_{A}(G^{2})) = 0, as claimed. Assume now that A is split. Then U_{A} is the orthogonal group of a quadratic form q; let sn : U_{A}(E_{A}) → H^{1}(E_{A}) be the associated spinor norm, and note that sn is a group homomorphism (see for instance [L 05], Chapter 5, Theorem 1.13). The homomorphism sn depends on the choice of the quadratic form q with orthogonal group U_{A}, but its restriction to U_{A}^{0} does not depend on this choice. Note that δ : U_{A}^{0}(E_{A}) → H^{1}(E_{A}) is the restriction of sn to U_{A}^{0}(E_{A}). Therefore for all g ∈ G, we have δ(i_{A}(g^{2})) = sn(i_{A}(g))^{2}, and since H^{1}(E_{A}) is a 2-torsion group, this implies that δ(i_{A}(G^{2})) = 0. This completes the proof of the lemma.

Let H be a subgroup of G^{2}. By Lemma 3.1, we have i_{A}(H) ⊂ s(Ũ_{A}(E_{A})). Let

V_{A}^{H} = Ũ_{A}(E_{A}) ×_{U_{A}^{0}(E_{A})} H

be the fibered product of s : Ũ_{A}(E_{A}) → U_{A}^{0}(E_{A}) and i_{A} : H → U_{A}^{0}(E_{A}). Therefore we have a central extension

1 → Z/2Z → V_{A}^{H} →^{p} H → 1,

where p is the projection to the factor H. Note that the surjectivity of p follows from the fact that by Lemma 3.1 every element of i_{A}(H) has a preimage in Ũ_{A}(E_{A}).

Let us denote by

e_{A}^{H} ∈ H^{2}(H)

the cohomology class corresponding to the extension V_{A}^{H}. If ϕ(Γ_{k}) ⊂ H, we denote by

ϕ^{∗} : H^{2}(H) → H^{2}(k)

the homomorphism induced by ϕ : Γ_{k} → H.

Proposition 3.2. Let ψ : Γ_{k} → G be another continuous homomorphism corresponding to the G–Galois algebra L. Set H_{ϕ} = ϕ(Γ_{k}) and H_{ψ} = ψ(Γ_{k}). Then we have

ϕ^{∗}(e_{A}^{H_{ϕ}}) = ψ^{∗}(e_{A}^{H_{ψ}}) in H^{2}(k).

Proof. We have ψ = Int(g) ∘ ϕ for some g ∈ G. Note that i_{A}(g) ∈ U_{A}(E_{A}), and that Int(i_{A}(g)) is an automorphism of U_{A}^{0}(E_{A}). Any automorphism of U_{A}^{0}(E_{A}) can be lifted to an automorphism of Ũ_{A}(E_{A}); indeed, such a lift exists over a separable closure, and is unique, hence defined over the ground field. Let f : Ũ_{A}(E_{A}) → Ũ_{A}(E_{A}) be a lift of Int(i_{A}(g)). Then f induces an isomorphism V_{A}^{H_{ϕ}} → V_{A}^{H_{ψ}}, which sends H_{ϕ} to H_{ψ}, and is the identity on Z/2Z. This implies that ϕ^{∗}(e_{A}^{H_{ϕ}}) = ψ^{∗}(e_{A}^{H_{ψ}}) in H^{2}(k).

The invariant c_{A}(L)

Recall that we assume that the H^{1}-condition is satisfied. We now choose for H the image ϕ(Γ_{k}) of Γ_{k} in G, and set V_{A} = V_{A}^{H}, e_{A} = e_{A}^{H}. We denote by c_{A}(L) the class of ϕ^{∗}(e_{A}) in H^{2}(k); Proposition 3.2 shows that this class does not depend on the choice of ϕ : Γ_{k} → G defining the G–Galois algebra L. Since H^{2}(k) ≃ Br_{2}(k), we can also consider c_{A}(L) as an element of Br_{2}(k).

Recall that the G-trace form q_{L} determines a rank one hermitian form over (A, σ_{A}). We want to relate c_{A}(L) to the Clifford invariant of this hermitian form.

The invariant clif_{A}(q_{L})

The map i_{A} : H → U_{A}^{0}(E_{A}) induces a map of pointed sets

i_{A} : H^{1}(E_{A}, H) → H^{1}(E_{A}, U_{A}^{0}).

Let u_{A}^{0}(L) be the image of [ϕ_{A}] ∈ H^{1}(E_{A}, H) by this map. Then the element u_{A}(L) defined in 1.5 is the image of u_{A}^{0}(L) under the further composition with the map H^{1}(E_{A}, U_{A}^{0}) → H^{1}(E_{A}, U_{A}).

Let us consider the exact sequence 1 → Z/2Z → Ũ_{A} → U_{A}^{0} → 1, and let δ be the connecting map H^{1}(E_{A}, U_{A}^{0}) → H^{2}(E_{A}) ≃ Br_{2}(E_{A}) of the associated cohomology exact sequence. Recall that ⟨A⟩ is the subgroup of Br_{2}(E_{A}) generated by the class of the algebra A. The Clifford invariant of q_{L} at A is by definition the image of δ(u_{A}^{0}(L)) in Br_{2}(E_{A})/⟨A⟩. Let us denote it by clif_{A}(q_{L}).

Theorem 3.3. The image of Res_{E_{A}/k}(c_{A}(L)) in Br_{2}(E_{A})/⟨A⟩ is equal to clif_{A}(q_{L}).

We need the following lemma :

Lemma 3.4. Let K be a field, let C be a finite group, and let f : Γ_{K} → C be a continuous homomorphism. Let us denote by [f] ∈ H^{1}(K, C) the corresponding cohomology class. Let

1 → Z/2Z → V → C → 1

be a central extension with trivial Γ_{K}-action. Let [e] ∈ H^{2}(C) be the class of a 2-cocycle e : C × C → Z/2Z representing this extension. Let ∂ : H^{1}(K, C) → H^{2}(K) be the connecting map associated to the above exact sequence, and let f^{∗} : H^{2}(C) → H^{2}(K) be the map induced by f. Then

f^{∗}([e]) = ∂([f]).

Proof. This follows from a direct computation. For all σ, τ ∈ Γ_{K}, we have f^{∗}(e)(σ, τ) = e(f(σ), f(τ)) = x_{f(σ)}x_{f(τ)}x_{f(στ)}^{−1}, where x : C → V is a section. On the other hand, (∂f)(σ, τ) = x_{f(σ)}^{f(σ)}(x_{f(τ)})x_{f(στ)}^{−1}, and this is equal to x_{f(σ)}x_{f(τ)}x_{f(στ)}^{−1}, since the action of Γ_{k} on V is trivial.

Proof of Theorem 3.3. Let ∂ : H^{1}(E_{A}, H) → H^{2}(E_{A}) be the connecting map of the cohomology exact sequence associated to the exact sequence

1 → Z/2Z → V_{A} → H → 1

with all the groups having trivial Γ_{E_{A}}-action. Recall that ϕ_{A} : Γ_{E_{A}} → Γ_{k} → H is the composition of ϕ : Γ_{k} → H with the inclusion of Γ_{E_{A}} into Γ_{k}. By Lemma 3.4 we have ∂([ϕ_{A}]) = ϕ_{A}^{∗}(e_{A}) = Res_{E_{A}/k}(ϕ^{∗}(e_{A})) = Res_{E_{A}/k}(c_{A}(L)). In view of the commutative diagram of Γ_{E_{A}}-groups

1 → Z/2Z → Ũ_{A}(k_{s}) → U_{A}^{0}(k_{s}) → 1
↑  ↑  ↑
1 → Z/2Z → V_{A} →  H  → 1

we have δ(u_{A}^{0}(L)) = ∂([ϕ_{A}]). Therefore we obtain Res_{E_{A}/k}(c_{A}(L)) = δ(u_{A}^{0}(L)). Since the class of δ(u_{A}^{0}(L)) in Br_{2}(E_{A})/⟨A⟩ is equal to clif_{A}(q_{L}) by definition, this completes the proof of the theorem.

Proposition 3.5. If L has a self-dual normal basis over k, then Res_{E_{A}/k}(c_{A}(L)) is trivial in Br_{2}(E_{A})/⟨A⟩.

Proof. Since L has a self-dual normal basis over k, the class u_{A}(L) corresponds to the class of the rank one unit hermitian form ⟨1⟩ in H^{1}(E_{A}, U_{A}). As ⟨1⟩ corresponds to the trivial cocycle in Z^{1}(E_{A}, U_{A}^{0}), its Clifford invariant is trivial, in other words, clif_{A}(q_{L}) is trivial. By Theorem 3.3 the image of Res_{E_{A}/k}(c_{A}(L)) in Br_{2}(E_{A})/⟨A⟩ is equal to clif_{A}(q_{L}), hence the proposition is proved.

We conclude this section with an example where c_{A}(L) ≠ 0, but Res_{E_{A}/k}(c_{A}(L)) = 0 (and hence clif_{A}(q_{L}) = 0) :

Example 3.6. Let G = A_{5}, the alternating group, and assume that k = Q. Let A be a factor of k[G] corresponding to a degree 3 orthogonal representation of G; then A = M_{3}(E_{A}) with E_{A} = k(\sqrt{5}), and the involution σ_{A} is induced by the unit form ⟨1, 1, 1⟩. Let ϵ ∈ G be a product of two disjoint transpositions.

Let z ∈ k^{×}, and let ψ : Γ_{k} → {1, ϵ} be the corresponding quadratic character. Let ϕ : Γ_{k} → G be given by ϕ = ι ∘ ψ, where ι : {1, ϵ} → G is the inclusion. Let L be the G-Galois algebra corresponding to ϕ. Set H = {1, ϵ}, and note that the image of ϕ is contained in H. Set N = k[X]/(X^{2} − z); then we have L = Ind_{H}^{G}(N).

Note that ϵ lifts to an element of order 4 in Ã_{5}, hence also in Ũ_{A}(E_{A}). Therefore the extension 1 → Z/2Z → V_{A}^{H} → H → 1 is not trivial; the group V_{A}^{H} is cyclic of order 4. Recall that e_{A} is the class of this extension in H^{2}(H); hence e_{A} is the only non-trivial element of H^{2}(H). By definition, we have c_{A}(L) = ϕ^{∗}(e_{A}), and this is equal to the cup product (z)(z) = (−1)(z) in H^{2}(k).

Set z = 11. Then c_{A}(L) = (−1)(11) is not trivial in H^{2}(k). On the other hand, since E_{A} = k(\sqrt{5}), we have Res_{E_{A}/k}(c_{A}(L)) = 0 in H^{2}(E_{A}). Note that the subgroup ⟨A⟩ of Br_{2}(E_{A}) is trivial, and recall that clif_{A}(q_{L}) = Res_{E_{A}/k}(c_{A}(L)) in Br_{2}(E_{A}) ≃ H^{2}(E_{A}) by Theorem 3.3; therefore we have clif_{A}(q_{L}) = 0.

§4. Unitary invariants

We keep the notation of the previous sections : G is a finite group, L is a G–Galois algebra, and ϕ : Γ_{k} → G is a homomorphism associated to L. We suppose that the H^{1}-condition is satisfied by ϕ : Γ_{k} → G, hence ϕ(Γ_{k}) is a subgroup of G^{2}. Let A be a unitary σ^{s}-stable central simple factor of k[G]^{s} (see 1.3). We denote by F_{A} be the center of A; note that F_{A} is a quadratic extension of E_{A}.

Using the same strategy as in §3, we first define an element of H^{2}(k) which is an invariant of the G-Galois algebra L. We then consider the hermitian form h_{A} over (A, σ) determined by q_{L}, and recall the definition of the discriminant of this form, thereby obtaining an element of Br_{2}(E_{A}). This is an invariant of the hermitian form h_{A}, and hence also of the G–form q_{L}. We then show that the restriction of the first invariant to H^{2}(E_{A}) is equal to the second one (see Theorem 4.5).

We start by recording some facts from Galois cohomology.

Let E be a field of characteristic ≠ 2, and let E_{s} be a separable closure of E. Let F be a quadratic extension of E, let x ↦ \overline{x} the non-trivial automorphism of F over E, and let F^{×1} be the subgroup of F^{×} consisting of the x ∈ F such that x\overline{x} = 1. Let N : F → E, given by N(x) = x\overline{x}, be the norm map. We denote by [F] the class of the quadratic extension F/E in H^{1}(E). For all x ∈ E^{×}, we denote by (x) the class of x in E^{×}/E^{×2}, and by [x] the class of x in E^{×}/N(F^{×}).

Lemma 4.1. (a) The connecting homomorphism E^{×} → H^{1}(E, R_{F/E}^{1}G_{m}) as-

sociated to the exact sequence 1 → R_{F/E}^{1}G_{m} → R_{F/E}G_{m} →^{N} G_{m} → 1 induces an isomorphism α : E^{×}/N(F^{×}) → H^{1}(E, R_{F/E}^{1}G_{m}).

(b) Let x ∈ E^{×}, and let f_{x} : Γ_{E} → R_{F/E}^{1}G_{m}(E_{s}) be defined by f_{x}(γ) = y^{−1}γ(y), where y ∈ (F ⊗_{E} E_{s})^{×} is such that N(y) = x. Then we have α((x)) = [f_{x}].

Proof. (a) follows from Hilbert’s theorem 90, and (b) from the definition of the connecting homomorphism.

¿From now on, we identify E^{×}/N(F^{×}) and H^{1}(E, R_{F/E}^{1}G_{m}) via the isomorphism α.

Lemma 4.2. Let 1 → Z/2Z → R_{F/E}^{1}G_{m} →^{s} R_{F/E}^{1}G_{m} → 1 be the exact sequence of linear algebraic groups with s the squaring map. Let δ : H^{1}(E, R_{F/E}^{1}G_{m}) → H^{2}(E) be the connecting homomorphism associated to this exact sequence. Identifying H^{1}(E, R_{F/E}^{1}G_{m}) with E^{×}/N(F^{×}) via α, we have

δ([x]) = (x)[F] ∈ H^{2}(E)

for all x ∈ E^{×}, where (x)[F] denotes the cup product of (x), [F] ∈ H^{1}(E).

Proof. A 2-cocycle associated to (x)[F] ∈ H^{2}(E) is given by f(σ, τ) such that f(σ, τ) = 1 if the restriction of σ to E(\sqrt{x}) is the identity, or if the restriction of τ to F is the identity, and f(σ, τ) = −1 otherwise. Let us check that the cohomology class of f in H^{2}(E) is equal to δ([x]). Let y ∈ (F ⊗_{E} E_{s})^{×} be such that N_{F ⊗_{E}E_{s}/E_{s}}(y) = y\overline{y} = x. A 1-cocycle g in Z^{1}(E, R_{F/E}^{1}G_{m}) associated to [x] is given by g(σ) = y^{−1}σ(y) for σ ∈ Γ_{E}. For all τ ∈ Γ_{E}, set z_{τ} = y^{−1}\sqrt{x} if the restriction of τ to F is not the identity, and z_{τ} = 1 otherwise. Then N_{F ⊗_{E}E_{s}/E_{s}}(z_{τ}) = z_{τ} \overline{z_{τ}} = (y^{−1}\sqrt{x})(\overline{y}^{−1}\sqrt{x}) if the restriction of τ to F is not the identity. Since y\overline{y} = x, we have z_{τ} ∈ R_{F/E}^{1}G_{m}(E_{s}). Further, s(z_{τ}) = y^{−2}x = y^{−1}τ(y) if the restriction of τ to F is not the identity, and s(z_{τ}) = 1 = y^{−1}τ(y) otherwise. Thus δ(g)(σ, τ) = z_{σ}^{σ}(z_{τ})z_{στ}^{−1}. It is straightforward to check that δ(g)(σ, τ) = 1 if the restriction of σ to E(\sqrt{x}) is the identity, or the restriction of τ to F is the identity, and that δ(g)(σ, τ) = −1 otherwise. This is precisely the cocycle f, hence we have δ([x]) = (x)[F] in H^{2}(E). This concludes the proof of the lemma.

Lemma 4.3. We have an injective homomorphism E^{×}/N(F^{×}) → Br_{2}(E) defined by [x] ↦ (x, F/E).

Proof. Indeed, the class of the quaternion algebra (x, F/E) is trivial in Br_{2}(E) if and only if x ∈ N(F^{×}).

We now define an invariant d_{A}(L) ∈ H^{2}(k, Z/2Z) of the G-Galois algebra L.

The invariant d_{A}(L)

Recall that F_{A}^{×1} is the subgroup of F_{A}^{×} consisting of the x ∈ F_{A} such that xσ_{A}(x) = 1; in other words, F_{A}^{×1} = R_{F_{A}/E_{A}}^{1}G_{m}(E_{A}). We denote by s : R_{F_{A}/E_{A}}^{1}G_{m} → R_{F_{A}/E_{A}}^{1}G_{m} the squaring map, and by n : U_{A} → R_{F_{A}/E_{A}}^{1}G_{m} the reduced norm. Recall that i_{A} : G → U_{A}(E_{A}) is the homomorphism defined in 1.5; we have n(i_{A}(G^{2})) ⊂ s(F_{A}^{×1}).

Let H be a subgroup of G^{2}. Let V_{A}^{H} = F_{A}^{×1} ×_{F_{A}^{×1}} H be the fibered product of s : F_{A}^{×1} → F_{A}^{×1} and n ∘ i_{A} : H → F_{A}^{×1}. Then the sequence

1 → Z/2Z → V_{A}^{H} → H → 1

is exact. Note that the surjectivity follows from the fact that n(i_{A}(H)) ⊂ s(F_{A}^{×1}). Therefore V_{A}^{H} is a central extension of H by Z/2Z. Recall that the H^{1}-condition implies that ϕ(Γ_{k}) ⊂ G^{2}.

Proposition 4.4. Let ψ : Γ_{k} → G be another continuous homomorphism corresponding to the G–Galois algebra L. Set H_{ϕ} = ϕ(Γ_{k}) and H_{ψ} = ψ(Γ_{k}). Then we have

ϕ^{∗}(e_{A}^{H_{ϕ}}) = ψ^{∗}(e_{A}^{H_{ψ}}) in H^{2}(k).

Proof. We have ψ = Int(g) ∘ ϕ for some g ∈ G. The map F_{A}^{×1} ×_{F^{×1}} H_{ϕ} → F_{A}^{×1} ×_{F^{×1}} H_{ψ}, given by (x, y) → (x, gyg^{−1}), gives rise to an isomorphism V_{A}^{H_{ϕ}} → V_{A}^{H_{ψ}} that is the identity on Z/2Z and sends H_{ϕ} to H_{ψ}. This implies that ϕ^{∗}(e_{A}^{H_{ϕ}}) = ψ^{∗}(e_{A}^{H_{ψ}}) in H^{2}(k).

We now choose for H the image ϕ(Γ_{k}) of Γ_{k} in G, and set V_{A} = V_{A}^{H}, e_{A} = e_{A}^{H}.

Notation. Let us denote by d_{A}(L) the class of ϕ^{∗}(e_{A}) in H^{2}(k); Proposition 4.4 shows that this class is independent of the choice of ϕ : Γ_{k} → G defining the G–Galois algebra L.

We define the discriminant of the G-form q_{L} at A, and compare it with the cohomology class d_{A}(L).

The invariant disc_{A}(q_{L})

Recall that composing ϕ_{A} : Γ_{E_{A}} → H with the map i_{A} : H → U_{A}(k_{s}) we obtain a 1-cocycle Γ_{E_{A}} → U_{A}(k_{s}), the class of which in H^{1}(E_{A}, U_{A}) is u_{A}(L). The reduced norm n : U_{A} → R_{F_{A}/E_{A}}^{1}G_{m} induces a map n : H^{1}(E_{A}, U_{A}) → E_{A}^{×}/N(F_{A}^{×}).

Notation. Set disc_{A}(q_{L}) = (n(u_{A}(L)), F_{A}/E_{A}) in Br_{2}(E_{A}).

Note that this is well–defined by Lemma 4.3. Since we have Br_{2}(E_{A}) ≃ H^{2}(E_{A}), we can also consider disc_{A}(q_{L}) as an element of H^{2}(E_{A}). Then disc_{A}(q_{L}) is given by the cup product n(u_{A}(L)).[F_{A}] in H^{2}(E_{A}). This invariant is related to the previously defined invariant d_{A}(L) as follows :

Theorem 4.5. We have disc_{A}(q_{L}) = Res_{E_{A}/k}(d_{A}(L)) in H^{2}(E_{A}).

Proof. Let ∂ : H^{1}(E_{A}, H) → H^{2}(E_{A}) be the connecting map of the exact sequence

1 → Z/2Z → V_{A} → H → 1

with all the groups having trivial Γ_{E_{A}}-action. By Lemma 3.4 we have

∂([ϕ_{A}]) = ϕ_{A}^{∗}(e_{A}) = Res_{E_{A}/k}(ϕ^{∗}(e_{A})) = Res_{E_{A}/k}(d_{A}(L)).

We have the commutative diagram

1 → Z/2Z →  V_{A}  →  H  → 1
↓  ↓  ↓
1 → Z/2Z → R_{F_{A}/E_{A}}^{1}G_{m}(k_{s}) →^{s} R_{F_{A}/E_{A}}^{1}G_{m}(k_{s}) → 1

where the second vertical map is the projection on the first factor, and the

third one is H →^{i_{A}} U_{A}(E_{A}) →^{n} R_{F_{A}/E_{A}}^{1}G_{m}(E_{A}).

Let δ : H^{1}(E, R_{F_{A}/E_{A}}^{1}G_{m}) → H^{2}(E_{A}) be the connecting homomorphism associated to the exact sequence

1 → Z/2Z → R_{F_{A}/E_{A}}^{1}G_{m} →^{s} R_{F_{A}/E_{A}}^{1}G_{m} → 1.

By the commutativity of the above diagram, we have δ([n(u_{A}(L))]) = ∂([ϕ_{A}]). Hence we have Res_{E_{A}/k}(d_{A}(L)) = δ([n(u_{A}(L)]). We have δ([n(u_{A}(L))]) = (n(u_{A}(L))).[F_{A}] by Lemma 4.2 and hence Res_{E_{A}/k}(d_{A}(L)) = disc_{A}(q_{L}), as claimed.

Lemma 4.6. If q_{L} corresponds to the hermitian form ⟨z_{A}⟩ over (A, σ_{A}), then we have

disc_{A}(q_{L}) = (n(z_{A}), F_{A}/E_{A}) in Br_{2}(E_{A}).

Proof. Set z = z_{A}. Let z = wσ_{A}(w) with w ∈ A ⊗_{E_{A}} k_{s}. The cocycle τ ↦ w^{−1}τ(w) represents the class of the hermitian form ⟨z⟩ in H^{1}(E_{A}, U_{A}). Let us denote this class by u_{z} ∈ H^{1}(E_{A}, U_{A}), and note that we have u_{z} = u_{A}(L) by definition. The cocycle τ ↦ n(w)^{−1}τ(n(w)) represents the class n(u_{z}) ∈ H^{1}(E_{A}, R_{F_{A}/E_{A}}^{1}G_{m}). By Lemma 4.1 this class is mapped by α^{−1} to [n(z)] ∈ E_{A}^{×}/N(F_{A}^{×}). Therefore we have (n(z), F_{A}/E_{A}) = (n(u_{A}(L)), F_{A}/E_{A}) = disc_{A}(q_{L}), as claimed.

Proposition 4.7. If L has a self-dual normal basis over k, then Res_{E_{A}/k}(d_{A}(L)) is trivial in Br_{2}(E_{A}).

Proof. Since L has a self-dual normal basis, q_{L} corresponds to the hermitian form ⟨1⟩ over (A, σ_{A}). By Lemma 4.6 this implies that disc_{A}(q_{L}) is trivial. Since by Theorem 4.5 we have disc_{A}(q_{L}) = Res_{E_{A}/k}(d_{A}(L)), the Proposition is proved.

Remark. There are examples where d_{A}(L) ≠ 0 but Res_{E_{A}/k}(d_{A}(L)) = 0 (hence also disc_{A}(q_{L}) = 0); see for instance Example 5.2 (i).

§5. Self-dual normal bases

We keep the notation of the previous sections. In particular, G is a finite group, L is a G-Galois algebra over k, and ϕ : Γ_{k} → G is a homomorphism associated to L. We now apply the results of the previous sections to give necessary conditions for the existence of a self-dual normal basis, and to show that these are also sufficient when k has cohomological dimension ≤ 2, see Proposition 5.1 and Theorem 5.3.

Putting together the results of §2 - §4, we have the following :

Proposition 5.1. Suppose that L has a self-dual normal basis over k. Then the H^{1}-condition is satisfied, and

(i) For all orthogonal σ^{s}-stable central simple factors A of k[G]^{s}, we have

Res_{E_{A}/k}(c_{A}(L)) = 0 in Br_{2}(E_{A})/⟨A⟩.

(ii) For all unitary σ^{s}-stable central simple factors A of k[G]^{s}, we have

Res_{E_{A}/k}(d_{A}(L)) = 0 in Br_{2}(E_{A}).

Proof. This follows from Propositions 2.1, 3.5 and 4.7.

Example 5.2. (i) The aim of this example is to reinterpret and complete Exemple 10.2 of [BSe 94] using the results of the present paper. Assume that G is cyclic of order 8, and let s be a generator of G; let ϵ = s^{4} be the element of order 2 of G. Let z ∈ k^{×}, and let σ : Γ_{k} → {1, ϵ} be the corresponding quadratic character. Let ϕ : Γ_{k} → G be given by ϕ = ι ∘ σ, where ι : {1, ϵ} → G is the inclusion. Let L be the G-Galois algebra corresponding to ϕ. Set H = {1, ϵ}, and note that the image of ϕ is contained in H. Set N = k[X]/(X^{2} − z); then we have L = Ind_{H}^{G}(N). Set A = k[X]/(X^{4} +1), and let us write k[G] = A′ ×A. It is easy to see that the image of H in A′ is trivial. The involution σ_{A} sends the class of X to the class of X^{−1}. If k contains the 4th roots of unity, then A is a product of two factors exchanged by the involution, hence there k[G] has no involution invariant factor in which the image of H is non trivial. In this case, L has a self-dual normal basis. Assume that k does not contain the 4th roots of unity. Then A is a field; we have F_{A} = A, and E_{A} = k[X]/(X^{2} − 2). Note that A is unitary. We have i_{A}(ϵ) = −1, hence i_{A}(H) = {1, −1}.

Let i ∈ F_{A} be a primitive 4th root of unity. By the definition of the extension

1 → Z/2Z → V_{A} → H → 1

(cf. §4), we see that V_{A} = {(1, 1), (−1, 1), (i, ϵ), (−i, ϵ)}, a cyclic group of order 4. Recall that e_{A} is the class of this extension in H^{2}(H); hence e_{A} is the only non-trivial element of H^{2}(H). We have d_{A}(L) = ϕ^{∗}(e_{A}) = (z, z) = (z, −1), and Res_{E_{A}/k}(d_{A}(L)) = (z, −1)_{E} = (z, F_{A}/E_{A}). Therefore we have

d_{A}(L) = 0 ⟺ z is a sum of two squares in k,

and

Res_{E_{A}/k}(d_{A}(L)) = 0 ⟺ z is a sum of two squares in E_{A} = k(\sqrt{2}).

It is easy to find examples where d_{A}(L) ≠ 0 and Res_{E_{A}/k}(d_{A}(L)) = 0; for instance, we can take k = Q and z = 3.

By Proposition 5.1 the existence of a self-dual normal basis implies that we have Res_{E_{A}/k}(d_{A}(L)) = 0. On the other hand, in [BSe 94], Exemple 10.2 it is checked by direct computation that if z is a sum of two squares in k(\sqrt{2}), then L has a self-dual normal basis. Hence we have

L has a self-dual normal basis over k ⟺ z is a sum of two squares in k(\sqrt{2}).

(ii) Assume that G = D_{4}, the dihedral group of order 8. Then a G-Galois algebra L has a self-dual normal basis if and only if either L is split or L = Ind_{H}^{G}(N) with H of order 2, and N = k[X]/(X^{2} − z) for some z ∈ k^{×} such that z is a sum of two squares in k.

Indeed, let ϕ : Γ_{k} → G be a homomorphism associated to L. Note that G^{2} is of order 2, hence the H^{1}-condition holds if and only if the image of ϕ is trivial, or equal to G^{2}; in other words, L is split, or induced from a G^{2}-Galois algebra. If L is split, then L has a self-dual normal basis. Set H = G^{2}, and assume that L = Ind_{H}^{G}(N), with N = k[X]/(X^{2} − z) for some z ∈ k^{×}. It remains to show that L has a self-dual normal basis if and only if z is a sum of two squares in k.

The group G has one degree 2 and four degree 1 orthogonal representations. Since the H^{1}-condition holds, the image of G is trivial in the factors of k[G] corresponding to the degree 1 representations. Let A = M_{2}(k), and let σ_{A} be the involution induced by the 2-dimensional unit form; then the factor of k[G] corresponding to the degree 2 orthogonal representation of G is equal to A.

Let q_{A}(L) be the 2-dimensional quadratic form corresponding to the cohomology class u_{A}(L). Note that L has a self-dual normal basis if and only if q_{A} ≃ ⟨1, 1⟩; this is equivalent with q_{A} having trivial determinant and trivial Hasse-Witt invariant. Recall that the H^{1}-condition is satisfied by hypothesis; hence we have u_{A}(L) ∈ H^{1}(k, U_{A}^{0}), and this implies that det(q_{A}(L)) = 1 in k^{×}/k^{×2}. Since A is a matrix algebra over k, we have w_{2}(q_{A}(L)) = clif(q_{A}(L)). By Theorem 3.3, this implies that w_{2}(q_{A}(L)) = c_{A}(L); hence it remains to prove that c_{A}(L) = 0 if and only if z is a sum of two squares in k.

If k contains the 4-th roots of unity, then U_{A}^{0} = Ũ_{A} = G_{m}. If k does not contain the 4-th roots of unity, then U_{A}^{0} = Ũ_{A} = R_{K/k}^{1}G_{m}, where K = k[X]/(X^{2} + 1).

In both cases, s : Ũ_{A} → U_{A}^{0} is the squaring map. Using this, we see that the extension 1 → Z/2Z → V_{A} → H → 1 is non-trivial, and that c_{A}(L) = (z, −1). Therefore c_{A}(L) = 0 if and only if z is a sum of two squares in k, and hence

L has a self-dual normal basis over k ⟺ z is a sum of two squares in k.

(iii) Let G = A_{4}, the alternating group of order 12, and assume for simplicity that char(k) ≠ 3 and that k contains the third roots of unity. Then k[G] = k ×k ×k ×M_{3}(k), where the first factor corresponds to the unit representation, the second and the third to the two representations of degree 1 with image of order 3, and the fourth one to the irreducible representation of degree 3. Let A = M_{3}(k) be the fourth factor, and note that the restriction of σ to A is induced by the 3-dimensional unit form. The extension 1 → Z/2Z → V_{A} → G → 1 defined in §3 is

1 → Z/2Z → Ã_{4} → A_{4} → 1,

corresponding to the unique non-trivial element e ∈ H^{2}(A_{4}) (see [Se 84], 2.3). Let L be a G-Galois algebra, and note that the H^{1}-condition is satisfied, since G has no quotient of order 2. Let E be the subalgebra of L fixed by the subgroup A_{3} of G = A_{4}; then E is an étale algebra of rank 4. Let ϕ : Γ_{k} → A_{4} be a homomorphism corresponding to L. By [Se 84], Theorem 1 we have ϕ^{∗}(e) = w_{2}(q_{E}), the Hasse-Witt invariant of the quadratic form q_{E}; hence the invariant c_{A}(L) is equal to w_{2}(q_{E}). Let q_{A}(L) be the 3-dimensional quadratic form corresponding to the cohomology class u_{A}(L). Then q_{E} ≃ q_{A}(L) ⊕ ⟨1⟩, and it is easy to check that q_{A}(L) ≃ ⟨1, 1, 1⟩ ⟺ w_{2}(q_{E}) = 0, hence u_{A}(L) = 0 ⟺ w_{2}(q_{E}) = 0. Therefore we have

L has a self-dual normal basis over k ⟺ w_{2}(q_{E}) = 0,

recovering a result of [BSe 94] (see [BSe 94], Exemple 1.6).

The case of cyclic groups of order a power of 2 is further developed in §6; we now look at fields of low cohomological dimension. Recall that the 2cohomological dimension of Γ_{k}, denoted by cd_{2}(Γ_{k}), is the smallest integer d such that H^{i}(k, C) = 0 for all i > d and for every finite 2-primary Γ_{k}-module C. For fields of cohomological dimension ≤ 1, the question of existence of self-dual normal bases is settled in [BSe 94], 2.2.

Theorem 5.3. Assume that cd_{2}(Γ_{k}) ≤ 2. Then L has a self-dual normal basis over k if and only if the H^{1}-condition is satisfied, and the conditions (i) and (ii) below hold :

(i) For all orthogonal σ^{s}-stable central simple factors A of k[G]^{s}, we have

Res_{E_{A}/k}(c_{A}(L)) = 0 in Br_{2}(E_{A})/⟨A⟩.

(ii) For all unitary σ^{s}-stable central simple factors A of k[G]^{s}, we have

Res_{E_{A}/k}(d_{A}(L)) = 0 in Br_{2}(E_{A}).

Proof. If L has a self-dual normal basis over k, then by Proposition 5.1 the H^{1}-condition, as well as conditions (i) and (ii) are satisfied. Conversely, let us assume that the H^{1}-condition, as well as conditions (i) and (ii) hold. Since the H^{1}-condition holds, we can define c_{A}(L) and d_{A}(L), cf. §3 and §4. By Theorems 3.3 and 4.5 we have clif_{A}(q_{L}) = Res_{E_{A}/k}(c_{A}(L)) and disc_{A}(q_{L}) = Res_{E_{A}/k}(d_{A}(L)). Therefore, conditions (i) and (ii) imply that clif_{A}(q_{L}) is trivial for all orthogonal factors A, and disc_{A}(q_{L}) is trivial for all unitary factors A. Let us prove that L has a self-dual normal basis over k. Let us denote by h_{A} the hermitian form over (A, σ_{A}) corresponding to u_{A}(L). It is enough to show that for all factors A, the class u_{A}(L) is trivial; this is equivalent with saying that the hermitian form h_{A} is isomorphic to the unit form 1_{A} over (A, σ_{A}). By Witt cancellation (see for instance [BPS 13], Theorem 2.5.2) this in turn is equivalent to saying that h_{A} ⊕ −1_{A} is hyperbolic. Let us prove this successively for symplectic, orthogonal and unitary characters.

Assume first that A is symplectic. Then by [BP 95], Theorem 4.3.1 every even dimensional non-degenerate hermitian form over a central simple algebra with involution is hyperbolic. This implies that h_{A} ⊕ −1_{A} is hyperbolic. Assume now that A is orthogonal, and note that the H^{1}-condition implies that u_{A}(L) is the image of a class u_{A}^{0}(L) of H^{1}(E_{A}, U_{A}^{0}). This implies that h_{A} has trivial discriminant. As we saw above, clif_{A}(q_{L}) is trivial, hence the form h_{A} ⊕ −1_{A} has trivial Clifford invariant. By [BP 95], Theorem 4.4.1 every even dimensional non-degenerate hermitian form over a central simple algebra having trivial discriminant and trivial Clifford invariant is hyperbolic, hence h_{A} ⊕ −1_{A} is hyperbolic. Assume finally that A is a unitary character. We have seen above that disc_{A}(q_{L}) is trivial, therefore the form h_{A} ⊕ −1_{A} has trivial discriminant. By [BP 95], Theorem 4.2.1 every even dimensional non-degenerate hermitian form over a central simple algebra having trivial discriminant is hyperbolic, hence h_{A} ⊕ −1_{A} is hyperbolic.

This implies that L has a self-dual normal basis over k, hence the theorem is proved.

Recall that ϕ : Γ_{k} → G is a homomorphism associated to the G-Galois algebra L, and that for all x ∈ H^{n}(G), we denote by x_{L} the image of x by ϕ^{∗} : H^{n}(G) → H^{n}(k). Let H = ϕ(Γ_{k}). For n = 2, we also need the image of x by the homomorphism ϕ^{∗} : H^{n}(H) → H^{n}(k); we denote this image by x_{L}^{H}.

Corollary 5.4. Assume that cd_{2}(Γ_{k}) ≤ 2, that the H^{1}-condition is satisfied, and that we have x_{L}^{H} = 0 for all x ∈ H^{2}(H). Then L has a self-dual normal basis over k.

Proof. This follows immediately from Theorem 5.3. Indeed, the H^{1}-condition is satisfied by hypothesis. Moreover, the classes c_{A}(L) and d_{A}(L) are by definition in the image of ϕ^{∗} : H^{2}(H) → H^{2}(k), hence the hypothesis x_{L}^{H} = 0 for all x ∈ H^{2}(H) implies that c_{A}(L) = 0 for all orthogonal factors A, and d_{A}(L) = 0 for all unitary factors A. Therefore conditions (i) and (ii) of Theorem 5.3 are satisfied, and hence L has a self-dual normal basis over k.

Remarks. (i) Corollary 5.4 suggests the following question : Assume that cd_{2}(Γ_{k}) ≤ 2, and that the H^{1}-condition is satisfied. If x_{L} = 0 for all x ∈ H^{2}(G), does it follow that L has a self-dual normal basis over k ? This follows from Corollary 5.4 when L is a field extension, in other words, when ϕ is surjective : indeed, then H = G.

(ii) The question above (see (i)) has a negative answer for fields of higher cohomological dimensions. Indeed, by [BSe 94], III. 10.1, there exist examples of G-Galois algebras L over fields of cohomological dimension 3 such that for all n > 0 we have x_{L} = 0 for all x ∈ H^{n}(G), but L does not have a self-dual normal basis over k.

(iii) The converse of the question raised in (i) also has a negative answer : indeed, there exist examples of G-Galois algebras L over Q having a self-dual normal basis such that there exists x ∈ H^{2}(G) with x_{L} ≠ 0 (see [BSe 94], III. 10.2).

The following result was proved in [BSe 94], Corollaire 3.2.2 in the case where k is an imaginary number field; the proof also applies for fields of cohomological dimension ≤ 2, using the results of [BP 95]. We give here an alternative proof.

Corollary 5.5. Assume that cd_{2}(Γ_{k}) ≤ 2, and that

H^{1}(G) = H^{2}(G) = 0.

Then L has a self-dual normal basis over k.

Proof. Since H^{1}(G) = 0, we have G = G^{2}. Let A be orthogonal or unitary, and let us construct a central extension V′_{A} of G by Z/2Z, as follows. If A is orthogonal, set V′_{A} = V_{A}^{G} = Ũ_{A}(E_{A}) ×_{U_{A}^{0}(E_{A})} G, with the notation of §3. If A is unitary, then we set V′_{A} = V_{A}^{G} = F_{A}^{×1} ×_{F_{A}^{×1}} G, the notation being as in §4. In each case, we get a central extension V′_{A} of G by Z/2Z. Since H^{2}(G) = 0, this extension is split. Note that the central extension V_{A} of H by Z/2Z constructed in §3 and §4 is a subgroup of V′_{A}, and that the restriction of the projection V′_{A} → G is the projection V_{A} → H. Hence the extension V_{A} is also split. This implies that we have c_{A}(L) = 0 for every orthogonal A, and d_{A}(L) = 0 for every unitary A. By Theorem 5.3 this implies that L has a self-dual normal basis over k.

§6. Cyclic groups of 2-power order

In this section, G is assumed to be cyclic of order 2^{n}, with n ≥ 2. We start by giving necessary and sufficient conditions for two G–Galois algebras to have isomorphic trace forms in terms of cohomological invariants of degree 1 and 2 (see Proposition 6.2), namely the degree 1 invariants introduced in [BSe 94], and the discriminants of the hermitian forms at the unitary factors (see §4).

We then use the invariants defined in the first part of §4 to give necessary and sufficient conditions for the existence of a self–dual normal basis. We start with settling the case where k contains the 4th roots of unity :

Proposition 6.1. Assume that k contains the 4th roots of unity. Let L and L′ be two G–Galois algebras. Then q_{L} ≃_{G} q_{L′} if and only if x_{L} = x_{L′} for all x ∈ H^{1}(G).

Proof. The algebra k[G] has two orthogonal factors k; since k contains the 4th roots of unity, there are no other involution invariant factors. Therefore u(L) = u(L′) if and only if the cohomology classes u associated to the two degree 1 orthogonal factors coincide, and this is equivalent with the condition x_{L} = x_{L′} for all x ∈ H^{1}(G). Hence, by [BSe 94], Proposition 1.5.1, we have q_{L} ≃_{G} q_{L′}.

More generally, we have :

Proposition 6.2. Let L and L′ be two G–Galois algebras. Then q_{L} ≃_{G} q_{L′} if and only if the following conditions hold :

(i) x_{L} = x_{L′} for all x ∈ H^{1}(G).

(ii) disc_{A}(q_{L}) = disc_{A}(q_{L′}) for all unitary factors A of k[G].

Before proving Proposition 6.2, note that when k contains the 4th roots of unity, then Proposition 6.2 follows from Proposition 6.1. Hence we only need to prove the proposition when k does not contain the 4th roots of unity.

From now on, we assume that k does not contain the 4th roots of unity. We start by introducing some notation. Set A(i) = k[X]/(X^{2^{i−1}} + 1), for i = 1, …, n; then the factors of k[G] are k, and A(1), …, A(n). Note that k and A(1) are orthogonal, and A(2), …, A(n) are unitary. For i = 2, …, n, we have A(i) = F_{A(i)}.

Proof of Proposition 6.2. Recall that we are assuming that k does not contain the 4th roots of unity (otherwise, the proposition follows from Proposition 6.1). For all factors A of k[G], let us denote by h_{A}, respectively h′_{A}, the hermitian form over (A, σ_{A}) determined by q_{L}, respectively q_{L′}.

Assume that q_{L} ≃_{G} q_{L′}. Then (i) holds by [BSe 94], Proposition 2.2.1. Let A be a unitary factor; then the hermitian forms h_{A} and h′_{A} are isomorphic. Since disc_{A}(q_{L}) and disc_{A}(q_{L′}) are invariants of these hermitian forms, condition (ii) holds as well.

Conversely, suppose that (i) and (ii) hold. Let us show that u_{A}(L) = u_{A}(L′) for all factors A. Condition (i) implies that this is true for A = k and A = A(1); indeed, in both cases the group U_{A} is of order 2. Let us assume that A is a unitary factor, that is, A = A(i) for some i = 2, …, n. Note that A = F_{A}, hence the hermitian forms h_{A} and h′_{A} are one dimensional hermitian forms over the commutative field F_{A}. Such a form is determined up to isomorphism by its discriminant; hence condition (ii) implies that h_{A} ≃ h′_{A}. Therefore we have u_{A}(L) = u_{A}(L′) for all factors A, hence u(L) = u(L′), and by [BSe 94], Proposition 1.5.1 we have q_{L} ≃_{G} q_{L′}. This completes the proof of the Proposition.

Let us recall a notation from [Se 84], 1.5 or [Se 92], 9.1.3 : if m is an integer, m ≥ 1, we denote by s_{m} ∈ H^{2}(S_{m}) the element of H^{2}(S_{m}) corresponding to the central extension

1 → Z/2Z → S̃_{m} → S_{m} → 1

which is characterized by the properties :

1. A transposition in S_{m} lifts to an element of order 2 in S̃_{m}.

2. A product of two disjoint transpositions lifts to an element of order 4 in S̃_{m}.

Note that s_{m} = 0 if and only if m ≤ 3 (see [Se 84], 1.5).

If m is a power of 2, m ≥ 2, let us denote by C_{m} the cyclic group of order m, and by e_{m} be the unique non-trivial element of H^{2}(C_{m}). Sending a generator of C_{m} to an m-cycle of S_{m} defines an injective homomorphism f : C_{m} → S_{m}; we denote by f^{∗} : H^{2}(S_{m}) → H^{2}(C_{m}) the homomorphism induced by f.

If q is a quadratic form over k, we denote by w_{2}(q) its Hasse-Witt invariant (see for instance [Se 84], 1.2 or [Se 92], 9.1.2); it is an element of H^{2}(k).

Lemma 6.3. Let m be a power of 2.

(i) We have f^{∗}(s_{m}) = e_{m} in H^{2}(C_{m}).

(ii) Let ψ : Γ_{k} → C_{m} be a continuous homomorphism, and let K be the étale algebra over k corresponding to ϕ. Then the obstruction to the lifting of ϕ to a homomorphism Γ_{k} → C_{2m} is

w_{2}(q_{K}) + (2)(D_{K})

where D_{K} is the discriminant of K, and (2)(D_{K}) denotes the cup product of the elements (2) and (D_{K}) of H^{1}(k).

Proof. (i) Let C̃_{m} be the inverse image of C_{m} in S̃_{m}; it suffices to show that C̃_{m} ≃ C_{2m}, in other words that C̃_{m} is a non-trivial extension of C_{m}. Raising an m-cycle of S_{m} to the \frac{m}{2}-th power yields a product of \frac{m}{2} disjoint transpositions, and the inverse image of such an element in S̃_{m} is of order 4. Hence C̃_{m} is a non-trivial extension of C_{m}.

(ii) The obstruction to the lifting of ψ is ψ^{∗}(e_{m}) ∈ H^{2}(k). Since f^{∗}(s_{m}) = e_{m} by (i), we have

(f ∘ ψ)^{∗}(s_{m}) = ψ^{∗}(e_{m}).

On the other hand, (f ∘ ψ)^{∗}(s_{m}) = w_{2}(q_{K}) + (2)(D_{K}) by [Se 84], Theorem 1.

Proposition 6.4. Let L be a G-Galois algebra, and assume that the H^{1}– condition holds. Then we have

(i) Let A be a unitary factor of k[G]. If A ≠ A(n), then d_{A}(L) = 0.

(ii) Let L = K × ⋯ × K, where K is a field extension of k. Then

d_{A(n)}(L) = w_{2}(q_{K}) + (2)(D_{K}).

Proof. Let ϕ : Γ_{k} → G be a homomorphism associated to L, let H = ϕ(Γ_{k}), and let us denote by |H| its order. Recall from §4 that the extension

(∗) 1 → Z/2Z → V_{A} → H → 1

is defined by V_{A} = {(x, h) ∈ F_{A}^{×1} × H | x^{2} = i_{A}(h)}. Let us show that this extension is split if A ≠ A(n). Note that the group V_{A} is abelian, and hence (∗) is not split if and only if V_{A} is a cyclic group of order 2|H|. On the other hand, if A ≠ A(n), then the order of i_{A}(H) is strictly less than |H|, hence the group V_{A} does not have any elements of order 2|H|. Therefore the extension (∗) is split, and hence d_{A}(L) = 0; this completes the proof of (i).

Let us prove (ii). If L is split, then (ii) obviously holds, hence we may assume that |H| ≥ 2. If A = A(n), then the group V_{A} is cyclic of order 2|H|, and the extension (∗) is not split. Recall that we denote by e_{A} ∈ H^{2}(H) the class of this extension, and that d_{A} = ϕ^{∗}(e_{A}) ∈ H^{2}(k). Note that ϕ^{∗}(e_{A}) is also the obstruction for the lifting of ϕ : Γ_{k} → H to a continuous homomorphism Γ_{k} → V_{A}; by Lemma 6.3 (ii) this obstruction is equal to w_{2}(q_{K}) + (2)(D_{K}), hence (ii) is proved.

Corollary 6.5. Let L be a G-Galois algebra, and assume that the H^{1}–condition holds. Then L has a self–dual normal basis if and only if Res_{E_{A(n)}/k}(d_{A(n)}(L)) = 0 in Br_{2}(E_{A(n)}).

Proof. Proposition 6.2 implies that L has a self-dual normal basis if and only if the H^{1}-condition holds and if disc_{A}(q_{L}) = 0 for all unitary factors A of k[G]. By Theorem 4.5 we have Res_{E_{A}/k}(d_{A}(L)) = disc_{A}(q_{L}), and Proposition 6.4 (i) implies that d_{A}(L) = 0 if A ≠ A(n). This completes the proof of the corollary.

Corollary 6.6. Let L be a G-Galois algebra, and assume that the H^{1}– condition holds. Let L = K × ⋯ × K, where K is a field extension of k, with Gal(K/k) cyclic of order m. If K can be embedded in a Galois extension of k with cyclic Galois group of order 2m, then L has a self-dual normal basis.

Proof. Assume that K can be embedded in a Galois extension of k with cyclic Galois group of order 2m. Then by Lemma 6.3 (ii) we have w_{2}(q_{K}) + (2)(D_{K}) = 0. By Proposition 6.4 (ii), this implies that d_{A(n)}(L) = 0, and hence by Corollary 6.5 the G-Galois algebra L has a self-dual normal basis.

Example 6.7. Assume that G is of order 8. Let a, b, c, ϵ ∈ k with a^{2}−b^{2}ϵ = c^{2}ϵ; assume c non-zero, and ϵ not a square. Set x = \sqrt{ϵ}, and let K = k(\sqrt{a + bx}); note that D_{K} = ϵ, and that K/k is a cyclic extension of degree 4 (see for instance [Se 92], Theorem 1.2.1). Let L be the G-Galois algebra induced from K. Let us prove that

L has a self-dual normal basis ⟺ a is a sum of two squares in k(\sqrt{2}).

Indeed, set A = A(3); by Corollary 6.5 the G-Galois algebra L has a selfdual normal basis if and only if Res_{E_{A}/k}(d_{A}(L)) = 0. We have d_{A}(L) = w_{2}(q_{K}) + (2)(ϵ) by Proposition 6.4 (ii).

Let us show that w_{2}(q_{K}) = (−1)(a). Set y = \sqrt{a + bx}. Then \{1, x, y, xy\} is a basis of K over k, and in this basis the quadratic form q_{K} is the orthogonal sum of the diagonal form ⟨1, ϵ⟩ and of the quadratic form q given by aX^{2} +2bϵXY + aϵY^{2}. The form q represents a, and its determinant is ϵ(a^{2} − b^{2}ϵ) = c^{2}ϵ^{2}, hence det(q) = 1 in k^{2}/k^{×2}. This implies that q ≃ ⟨a, a⟩, hence q_{K} ≃ ⟨1, ϵ, a, a⟩, and w_{2}(q_{K}) = (a)(a) = (−1)(a).

Therefore d_{A}(L) = (−1)(a) + (2)(ϵ). Note that E_{A} = k(\sqrt{2}); hence Res_{A/k}(d_{A}(L)) = Res_{k(\sqrt{2})/k}((−1)(a)), and this element is 0 if and only if a is a sum of two squares in k(\sqrt{2}).

Note that combining this example with Example 5.2 (i) we get a necessary and sufficient condition for a C_{8}-Galois algebra to have a self-dual normal basis.

§7. Self-dual normal bases over local fields

We keep the notation of the previous sections, and assume that k is a (nonarchimedean) local field. The aim of this section is to give a necessary and sufficient condition for the existence of self-dual normal bases in terms of invariants defined over k.

We say that A is split if it is a matrix algebra over its center.

Theorem 7.1. The G-Galois algebra L has a self-dual normal basis if and only if the H^{1}-condition holds, and

(i) For all orthogonal A such that [E_{A} : k] is odd and A is split, we have c_{A}(L) = 0 in Br_{2}(k).

(ii) For all unitary A such that [E_{A} : k] is odd, we have d_{A}(L) = 0 in Br_{2}(k).

Proof. Assume that the H^{1}-condition is satisfied and that (i) and (ii) hold. Note that if A is not split, then we have Br_{2}(E_{A})/⟨A⟩ = 0, and that if [E_{A} : k] is even, then the map Res_{E_{A}/k} : Br_{2}(k) → Br_{2}(E_{A}) is trivial. Therefore for all orthogonal A we have Res_{E_{A}/k}(c_{A}(L)) = 0 in Br_{2}(E_{A})/⟨A⟩, and for all unitary A we have Res_{E_{A}/k}(d_{A}(L)) = 0 in Br_{2}(E_{A}). By Theorem 5.3, this implies that L has a self-dual normal basis.

Conversely, suppose that L has a self-dual normal basis. Then the H^{1}-condition holds by Proposition 2.1. By Theorem 5.1 we have Res_{E_{A}/k}(c_{A}(L)) = 0 in Br_{2}(E_{A})/⟨A⟩ for all orthogonal A. Since Res_{E_{A}/k} : Br_{2}(k) → Br_{2}(E_{A}) is injective if [E_{A} : k] is odd, condition (i) holds. Moreover, Theorem 5.1 implies that if A is unitary, then Res_{E_{A}/k}(d_{A}(L)) = 0 in Br_{2}(E_{A}). Applying again the injectivity of Res_{E_{A}/k} when [E_{A} : k] is odd, we obtain condition (ii). This completes the proof of the theorem.

§8. Self-dual normal bases over global fields

We keep the notation of the previous sections. Assume that k is a global field, and let Ω_{k} be the set of places of k. For all v ∈ Ω_{k}, we denote by k_{v} the completion of k at v. For all k-algebras R, set R^{v} = R ⊗_{k} k_{v}. We say that a G-Galois algebra is split if it is isomorphic to a direct product of copies of k permuted by G. We now apply the Hasse principle of [BPS 13] together with Theorem 7.1 above to give necessary and sufficient conditions for the existence of a self-dual normal basis over k.

Note that the fields E_{A} are abelian Galois extensions of k (cf. 1.2).

For all finite places v, let us write E_{A}^{v} = K_{A}(v) × ⋯ × K_{A}(v), where K_{A}(v) is a field extension of k_{v}. Set n_{A}^{v} = [K_{A}(v) : k_{v}].

We need additional notation in the case when A is unitary. Note that while A is a central simple algebra over F_{A}, and F_{A}/E_{A} is a quadratic extension, for some places v ∈ Ω_{k} we may have F_{A}^{v} = E_{A}^{v} × E_{A}^{v} with σ_{A} permuting the components, and A^{v} = B × B for some k_{v}-algebra B. In order to take this into account, we set ϵ_{A}^{v} = 0 if F_{A}^{v} = E_{A}^{v} × E_{A}^{v}, and ϵ_{A}^{v} = 1 otherwise.

Theorem 8.1. The G-Galois algebra L has a self-dual normal basis if and only if the H^{1}-condition holds, if L^{v} is split for all real places v, and if for all finite places v we have

(i) For all orthogonal A such that n_{A}^{v} is odd and A^{v} is split, we have c_{A}(L) = 0 in Br_{2}(k_{v}).

(ii) For all unitary A such that n_{A}^{v} is odd and ϵ_{A}^{v} = 1, we have d_{A}(L) = 0 in Br_{2}(k_{v}).

Proof. If L has a self-dual normal basis, then L^{v} is split for all real places v by [BSe 94], Corollaire 3.1.2, and conditions (i) and (ii) hold for all finite places v by Theorem 7.1. Conversely, assume that L^{v} is split for all real places v, and that for all finite places v conditions (i) and (ii) hold. Then [BSe 94], Corollary 3.1.2 (for real places) and Theorem 7.1 (for finite places) imply the existence of a self-dual normal basis for L^{v}, for all v ∈ Ω_{k}. By the Hasse principle result of [BPS 13], Theorem 1.3.1, the G-Galois algebra L has a self-dual normal basis over k.

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E. Bayer-Fluckiger École Polytechnique Fédérale de Lausanne EPFL/FSB/MATHGEOM/CSAG Station 8 1015 Lausanne, Switzerland eva.bayer@epfl.ch R. Parimala Department of Mathematics & Computer Science Emory University Atlanta, GA 30322, USA. parimala@mathcs.emory.edu