TRANSFER OF QUADRATIC FORMS AND OF QUATERNION ALGEBRAS OVER QUADRATIC FIELD

EXTENSIONS

KARIM JOHANNES BECHER, NICOLAS GRENIER-BOLEY, AND JEAN-PIERRE TIGNOL

A well-known theorem of Albert states that if a tensor product of two quaternion division algebras Q_{1}, Q_{2} over a field F of characteristic different from 2 is not a division algebra, then there exists a quadratic extension L of F that embeds as a subfield in Q_{1} and in Q_{2}; see [6, (16.29)]. The same property holds in characteristic 2, with the additional condition that L/F is separable: this was proved by Draxl [2], and several proofs have been proposed: see [3, Th. 98.19], and [7] for a list of earlier references.

Our purpose in this note is to extend the Albert–Draxl Theorem by substituting for the tensor product of two quaternion algebras the corestriction of a single quaternion algebra over a quadratic extension. Our main result is the following:

0.1. Theorem. Let F be an arbitrary field and let K be a quadratic étale F -algebra.^{1} For every quaternion K-algebra Q, the following conditions are equivalent:

(i) Q contains a quadratic F -algebra linearly disjoint from K; (ii) Q contains a quadratic étale F -algebra linearly disjoint from K; (iii) Cor_{K/F} Q is not a division algebra.

Note that when K = F × F the quaternion K-algebra Q has the form Q_{1}×Q_{2} for some quaternion F -algebras Q_{1}, Q_{2}, and Cor_{K/F} Q = Q_{1}⊗_{F} Q_{2}. Thus, in this particular case Theorem 0.1 is equivalent to the Albert–Draxl Theorem. The more general case is needed for the proof of the main result in [1].

If the characteristic char F is different from 2, Theorem 0.1 is proved in [6, (16.28)]. The proof below is close to that in [6], but it does not require any restriction on the characteristic. The idea is to use a transfer of the norm form n_{Q} of Q to obtain an Albert form of Cor_{K/F} Q, which allows us to substitute for (iii) the condition that the transfer of n_{Q} has Witt index at least 2. To complete the argument, we need to relate totally isotropic subspaces of the transfer to subforms of n_{Q} defined over F. This is slightly more delicate in characteristic 2. Therefore, we first discuss the transfer of quadratic forms in §1, and give the proof of Theorem 0.1 in §2. In the last section, we sketch an alternative proof of Theorem 0.1 based on a proof of the Albert–Draxl Theorem due to Knus [5]. This alternative proof relies on an explicit construction of an Albert form for the corestriction of a quaternion algebra.

The first author was supported by the FWO Odysseus Programme (project Explicit Methods in Quadratic Form Theory), funded by the Fonds Wetenschappelijk Onderzoek – Vlaanderen. The third author acknowledges support from the Fonds de la Recherche Scientifique–FNRS under grant n^{∘} J.0014.15.

1This means that K is either a quadratic separable field extension of F, or K ≃ F ×F.

Notations and Terminology. Quaternion algebras over an arbitrary field F are F -algebras obtained from an étale quadratic F -algebra E and an element a ∈ F^{×} by the following construction:

(E/F, a) = E ⊕ Ez

with multiplication defined by the equations

z^{2} = a  and  zℓ = ι(ℓ)z for ℓ ∈ L,

where ι is the nontrivial automorphism of L.

For quadratic and bilinear forms, we generally follow the notation and terminology of [3]. Thus, if φ: V → F is a quadratic form on a (finitedimensional) vector space V over an arbitrary field F, we let 𝔟_{φ}: V ×V → F denote the polar form of φ, defined by

𝔟_{φ}(x, y) = φ(x + y) − φ(x) − φ(y) for x, y ∈ V.

We set

rad 𝔟_{φ} = {x ∈ V | 𝔟_{φ}(x, y) = 0 for all y ∈ V }
rad φ = {x ∈ rad 𝔟_{φ} | φ(x) = 0}

and observe that these sets are F -subspaces of V with rad φ ⊆ rad 𝔟_{φ}. Moreover, if char F ≠ 2 then φ(x) = \frac{1}{2}𝔟_{φ}(x, x) for all x ∈ V and thus rad φ = rad 𝔟_{φ}. We call the quadratic form φ nonsingular^{2} if rad 𝔟_{φ} = {0}, regular if rad φ = {0} and nondegenerate if φ_{K} is regular for every field extension K/F or equivalently (by [3, Lemma 7.16]) if φ is regular and dim_{F} rad 𝔟_{φ} ≤ 1. Thus, every nonsingular form is nondegenerate and every nondegenerate form is regular; moreover, all three conditions are equivalent in the case where char F ≠ 2.

The Witt index of a quadratic form φ on a vector space V is the dimension of the maximal totally isotropic subspaces of V, i.e., the maximal subspaces U ⊆ V such that φ(u) = 0 for all u ∈ U; see [3, Prop. 8.11]. We write 𝔦_{0}(φ) for the Witt index of φ.

We will need the following easy observation:

2Nonsingular quadratic forms are not defined in [3].

0.2. Lemma. Let φ be a regular quadratic form on a vector space V. If φ is isotropic, the isotropic vectors span V.

Proof. Let V_{0} ⊆ V be the subspace spanned by the isotropic vectors of V, and let v ∈ V \ {0} be an isotropic vector. Since rad(φ) = {0} there exists w ∈ V such that 𝔟_{φ}(v, w) = 1. If x ∈ V is such that 𝔟_{φ}(v, x) ≠ 0, then the vector x − φ(x)𝔟_{φ}(v, x)^{−1}v is isotropic, hence it belongs to V_{0}. It follows that x ∈ V_{0} since v ∈ V_{0}. Thus, V_{0} contains all the vectors that are not orthogonal to v. In particular, it contains w. If x ∈ V is orthogonal to v, then 𝔟_{φ}(v, x+w) = 1, hence x+w ∈ V_{0}, and therefore x ∈ V_{0} since w ∈ V_{0}. Thus, V_{0} = V. □

1. Isotropic transfers

Let F be an arbitrary field and let K be a quadratic field extension of F. Fix a nonzero F -linear functional s: K → F such that s(1) = 0. For every quadratic form φ: V → K on a K-vector space, the transfer s_{∗}φ is the quadratic form on V (viewed as an F -vector space) defined by

s_{∗}φ(x) = s(φ(x)) for x ∈ V.

If φ is nonsingular, then s_{∗}φ is nonsingular: see [3, Lemma 20.4]. For every quadratic form ψ over F, we let ψ_{K} denote the quadratic form over K obtained from ψ by extending scalars to K.

The following result is well-known in characteristic different from 2, but it appears to be new in characteristic 2.

1.1. Theorem. Let φ be a nonsingular quadratic form over K. Then there exists a nondegenerate quadratic form ψ over F with dim ψ = 𝔦_{0}(s_{∗}φ) such that ψ_{K} is a subform of φ.

Proof. Substituting for φ its anisotropic part, we may assume φ is anisotropic. We may also assume 𝔦_{0}(s_{∗}φ) ≥ 1, for otherwise there is nothing to show. Pick an isotropic vector u ∈ V for the form s_{∗}φ; we thus have φ(u) ∈ F and φ(u) ≠ 0 since φ is anisotropic. If 𝔦_{0}(s_{∗}φ) = 1, then we may choose ψ = φ|_{F v}. For the rest of the proof, we assume 𝔦_{0}(s_{∗}φ) ≥ 2, and we argue by induction on 𝔦_{0}(s_{∗}φ).

Since φ is nonsingular, we may find v ∈ V such that 𝔟_{φ}(u, v) = 1. Let λ ∈ K be such that s(λ) = 1. We have s_{∗}φ(u) = 0 and

𝔟_{s_{∗}φ}(u, λv) = s(𝔟_{φ}(u, λv)) = s(λ) = 1,

hence the restriction of s_{∗}φ to the F -subspace U ⊆ V spanned by u and λv is nonsingular and isotropic. Therefore, s_{∗}φ|_{U} is hyperbolic. Let W = U^{⊥} ⊆ V be the orthogonal complement of U in V for the form s_{∗}φ. Since 𝔦_{0}(s_{∗}φ) ≥ 2, the form s_{∗}φ|_{W} is isotropic. By Lemma 0.2 we may find an isotropic vector w ∈ W such that 𝔟_{φ}(u, w) ≠ 0. However, 𝔟_{s_{∗}φ}(u, w) = 0 since w ∈ W = U^{⊥}; therefore 𝔟_{φ}(u, w) ∈ F^{×}. Moreover φ(w) ∈ F since w is isotropic for s_{∗}φ. Therefore, the restriction of φ to the F -subspace of V spanned by u and w is a quadratic form ψ_{1} over F.

Observe that u and w are K-linearly independent: if w = αu with α ∈ K, then

𝔟_{φ}(u, w) = α𝔟_{φ}(u, u) = 2αφ(u).

Since 𝔟_{φ}(u, w) ∈ F^{×} and φ(u) ∈ F, it follows that α ∈ F, which is impossible. Therefore, ψ_{1K} is a 2-dimensional subform of φ, and ψ_{1} is nonsingular because φ is anisotropic. Since s_{∗}(ψ_{1K}) is hyperbolic, for the orthogonal complement φ′ of ψ_{1K} in φ we obtain 𝔦_{0}(s_{∗}φ′) = 𝔦_{0}(s_{∗}φ) − 2. The theorem follows by induction. □

Remarks. 1. If s_{∗}φ is hyperbolic, then 𝔦_{0}(s_{∗}φ) = dim φ, hence Theorem 1.1 shows that φ = ψ_{K} for some quadratic form ψ over F. This particular case of Theorem 1.1 is established in [3, Th. 34.9].

2. If char F = 2 and 𝔦_{0}(s_{∗}φ) is odd, then the quadratic form ψ in Theorem 1.1 cannot be nonsingular, since nonsingular quadratic forms in characteristic 2 are even-dimensional; in particular, ψ_{K} is not an orthogonal direct summand of φ.

3. If the extension K/F is purely inseparable, then 𝔦_{0}(s_{∗}φ) is necessarily even. This follows because the K-subspace spanned by each isotropic vector for s_{∗}φ is a 2-dimensional F -subspace that is totally isotropic for s_{∗}φ.

4. The analogue of Theorem 1.1 for symmetric bilinear forms is proved in [3, Prop. 34.1].

2. Proof of Theorem 0.1

Let F be an arbitrary field. As in [3], we write I_{q}(F) for the Witt group of nonsingular quadratic forms of even dimension over F, and I(F) for the ideal of even-dimensional forms in the Witt ring W (F) of nondegenerate symmetric bilinear forms over F, and we let I_{q}^{n}(F) = I^{n−1}(F)I_{q}(F) for n ≥ 2. Let also Br_{2}(F) denote the 2-torsion subgroup of the Brauer group of F. Recall from [3, Th. 14.3] the group homomorphism

e_{2} : I_{q}^{2}(F) → Br_{2}(F)

defined by mapping the Witt class of a quadratic form φ to the Brauer class of its Clifford algebra.

2.1. Lemma. Let K be a quadratic field extension of an arbitrary field F, and let s: K → F be a nonzero F -linear functional such that s(1) = 0. The following diagram is commutative:

I_{q}^{2}(K) ⟶^{s_{∗}} I_{q}^{2}(F)
e_{2}↓  ↓^{e_{2}}
Br_{2}(K) ⟶^{Cor_{K/F}} Br_{2}(F).

Following the definition in [4, Rem. 6.9.2], Cor_{K/F} : Br_{2}(K) → Br_{2}(F) is the zero map if K is a purely inseparable extension of F.

Proof. We have I_{q}^{2}(K) = I(F)I_{q}(K) + I(K)I_{q}(F) by [3, Lemma 34.16], hence I_{q}^{2}(K) is generated by Witt classes of 2-fold Pfister forms that have a slot in F. Commutativity of the diagram follows by Frobenius reciprocity [3, Prop. 20.2], the computation of transfers of 1-fold Pfister forms in [3, Lemma 34.14] and [3, Cor. 34.19], and the projection formula in cohomology [4, Prop. 3.4.10]. □

Proof of Theorem 0.1. Since (ii) ⇒ (i) is clear, it suffices to prove (i) ⇒ (iii) and (iii) ⇒ (ii).

If (i) holds, then we may represent Q in the form (LK/K, b) where L is a quadratic étale F -algebra linearly disjoint from K and b ∈ K^{×}, or in the form (M/K, b) where M is a quadratic étale K-algebra and b ∈ F^{×}. In each case the projection formula [4, Prop. 3.4.10] shows that Cor_{K/F} Q is Brauer-equivalent to a quaternion algebra, hence (iii) holds.

Now, assume (iii) holds. If Q is split, then it contains an F -algebra isomorphic to F × F, so (ii) holds. For the rest of the proof, we assume Q is a division algebra. Let n_{Q} be the norm form of Q, which is a 2-fold Pfister quadratic form in I_{q}^{2}(K) such that e_{2}(n_{Q}) = Q in Br(K). Since n_{Q} represents 1, the transfer s_{∗}(n_{Q}) is isotropic, hence Witt-equivalent to a 6-dimensional nonsingular quadratic form φ in I_{q}^{2}(F). This form satisfies e_{2}(φ) = Cor_{K/F} (Q) in Br(F), hence φ is an Albert form of Cor_{K/F} (Q) as per the definition in [6, (16.3)]. In particular, since Cor_{K/F} (Q) is not a division algebra, φ is isotropic by [6, (16.5)], and therefore 𝔦_{0}(s_{∗}(n_{Q})) ≥ 2. By Theorem 1.1 there exists a nonsingular quadratic form ψ over F with dim ψ = 2 such that ψ_{K} is a subform of n_{Q}. Since Q is a division algebra, we have that ψ_{K} is anisotropic, hence ψ is similar to the norm form of a unique separable quadratic field extension L/F. The field L is linearly disjoint from K over F because ψ_{K} is anisotropic. On the other hand, ψ_{KL} is hyperbolic, hence KL splits the form n_{Q}, and it follows that there exists a K-algebra embedding of KL in Q. Therefore, (ii) holds. □

If K is a purely inseparable quadratic extension of F, all the statements of Theorem 0.1 hold for every quaternion algebra over K. To see this, recall from the definition of Cor_{K/F} in [4, Rem. 6.9.2] that the corestriction of every quaternion K-algebra is split. Moreover, if Q = (M/K, b) with M a separable quadratic extension of K, then the separable closure of F in M is a separable quadratic extension of F contained in Q and linearly disjoint from K.

3. The Albert form of a corestriction

Let Q be a quaternion algebra over a separable quadratic field extension K of an arbitrary field F. By definition (see [6, (16.3)]), the Albert forms of Cor_{K/F} Q are the 6-dimensional nonsingular quadratic forms in I_{q}^{2}(F) such that e_{2}(φ) = Cor_{K/F} Q in Br_{2}(F); they are all similar. As observed in the proof of Theorem 0.1, an Albert form of Cor_{K/F} Q may be obtained from the Witt class of the (8-dimensional) transfer s_{∗}(n_{Q}) of the norm form of Q for an arbitrary nonzero F -linear functional s: K → F such that s(1) = 0. In this section, we sketch a more explicit construction of an Albert form of Cor_{K/F} Q, inspired by Knus’s proof of the Albert–Draxl Theorem in [5], and we use it to give an alternative proof of Theorem 0.1.

We first recall the construction of the corestriction Cor_{K/F} Q. Let γ be the nontrivial F -automorphism of K and let ^{γ}Q denote the conjugate quaternion algebra ^{γ}Q = {^{γ}x | x ∈ Q} with the operations

γx +^{γ}y = ^{γ}(x + y), ^{γ}x ·^{γ}y = ^{γ}(xy), λ ·^{γ}x = ^{γ}(γ(λ)x)

for x, y ∈ Q and λ ∈ K. The algebra ^{γ}Q ⊗_{K} Q carries a γ-semilinear automorphism s defined by

s(^{γ}x ⊗ y) = ^{γ}y ⊗ x for x, y ∈ Q.

By definition, the corestriction (or norm) Cor_{K/F} (Q) is the F -algebra of fixed points (see [6, (3.12)]):

Cor_{K/F} (Q) = (^{γ}Q ⊗_{K} Q)^{s}.

Let Trd and Nrd denote the reduced trace and the reduced norm on Q. Let also σ be the canonical (conjugation) involution on Q. Consider the following K-subspace of ^{γ}Q ⊗_{K} Q:

V = {^{γ}x_{1} ⊗ 1 − 1 ⊗ x_{2} | x_{1}, x_{2} ∈ Q and γ(Trd(x_{1})) = Trd(x_{2})}.

This K-vector space has dimension 6 and is preserved by s, and one can show that the F -space of s-invariant elements has the following description, where T_{K/F} : K → F is the trace form:

V^{s} = {^{γ}y ⊗ 1 + 1 ⊗ y | y ∈ Q and T_{K/F} (Trd(y)) = 0}.

Now, pick an element κ ∈ K^{×} such that γ(κ) = −κ. (If char F = 2 we may pick κ = 1.) The following formula defines a quadratic form φ: V^{s} → F: for y ∈ Q such that T_{K/F} (Trd(y)) = 0, let

φ(^{γ}y ⊗ 1 + 1 ⊗ y) = κ · (γ(Nrd(y)) − Nrd(y)).

Nonsingularity of the form φ is easily checked after scalar extension to an algebraic closure of F, and computation shows that the linear map

f: V^{s} → M_{2}(Cor_{K/F} (Q)) given by ξ ↦ (\begin{matrix}0 & κ · (σ ⊗ id)(ξ) \\ ξ & 0 \end{matrix})

satisfies f(ξ)^{2} = φ(ξ) for all ξ ∈ V^{s}. Therefore, f induces an F -algebra homomorphism f_{∗} defined on the Clifford algebra C(V^{s}, φ). Dimension count shows that f_{∗} is an isomorphism

(3.1)  f_{∗}: C(V^{s}, φ) ⟶^{∼} M_{2}(Cor_{K/F} Q).

The restriction to the even Clifford algebra is an isomorphism C_{0}(V^{s}, φ) ≃ (Cor_{K/F} Q) × (Cor_{K/F} Q), hence the discriminant (or Arf invariant) of φ is trivial. This means φ ∈ I_{q}^{2}(F), and (3.1) shows that e_{2}(φ) = Cor_{K/F} Q in Br_{2}(F), so φ is an Albert form of Cor_{K/F} Q.

We use the Albert form φ to sketch an alternative proof of Theorem 0.1. Since all the conditions in Theorem 0.1 trivially hold if Q is split, we may assume Q is a division algebra. In particular, the base field F is infinite.

Suppose condition (i) of Theorem 0.1 holds. If x ∈ Q generates a quadratic F -algebra disjoint from K, then Trd(x) ∈ F and Nrd(x) ∈ F (and x ∉ K), hence ^{γ}(κx) ⊗ 1 + 1 ⊗ (κx) ∈ V^{s} is an isotropic vector of φ. Since φ is an Albert form of Cor_{K/F} Q, it follows that Cor_{K/F} Q is not a division algebra. Therefore, (i) implies (iii).

For the converse, suppose (iii) holds, and let ^{γ}y ⊗ 1 + 1 ⊗ y ∈ V^{s} be an isotropic vector for φ. Then γ(Nrd(y)) − Nrd(y) = 0, hence Nrd(y) ∈ F, and T_{K/F} (Trd(y)) = 0. Assuming y ∈ K quickly leads to a contradiction, and a density argument shows that we may find such an element y with Trd(y) ≠ 0. Then κy ∈ Q satisfies

Trd(κy) ∈ F, Trd(κy) ≠ 0, and Nrd(κy) = κ^{2} Nrd(y) ∈ F.

Therefore, κy generates a quadratic étale F -subalgebra of Q linearly disjoint from K, proving that (ii) (hence also (i)) holds.

References

[1] K.-J. Becher, N. Grenier-Boley, J.-P. Tignol, Involutions and stable subalgebras, in preparation.

[2] P. Draxl, Über gemeinsame separabel-quadratische Zerfällungskörper von Quaternionenalgebren, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1975, no. 16, 251–259.

[3] R. Elman, N. Karpenko, A. Merkurjev. The algebraic and geometric theory of quadratic forms. American Mathematical Society Colloquium Publications, 56, Amer. Math. Soc., Providence, RI, 2008.

[4] P. Gille and T. Szamuely, Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathematics, 101, Cambridge Univ. Press, Cambridge, 2006.

[5] M.-A. Knus, Sur la forme d’Albert et le produit tensoriel de deux algèbres de quaternions, Bull. Soc. Math. Belg. Sér. B 45 (1993), no. 3, 333–337.

[6] M.-A. Knus, A. S. Merkurjev, M. Rost, and J.-P. Tignol. The book of involutions. American Mathematical Society Colloquium Publications 44. American Mathematical Society, Providence, RI, 1998.

[7] J. Tits, Sur les produits tensoriels de deux algèbres de quaternions, Bull. Soc. Math. Belg. Sér. B 45 (1993), no. 3, 329–331.

Universiteit Antwerpen, Departement Wiskunde en Informatica, Middelheimlaan 1, B-2020 Antwerpen, Belgium

E-mail address: KarimJohannes.Becher@uantwerpen.be

Université de Rouen Normandie, Laboratoire de Didactique André Revuz (LDAR-EA 4434), F-76130 Mont-Saint-Aignan, France.

E-mail address: nicolas.grenier-boley@univ-rouen.fr

Université catholique de Louvain, ICTEAM Institute, Avenue G. Lemaître 4, Box L4.05.01, B-1348 Louvain-La-Neuve, Belgium.

E-mail address: jean-pierre.tignol@uclouvain.be