On the hermitian u-invariant of a

Karim Johannes Becher ∗, Mohammad G. Mahmoudi †

Abstract The u-invariant of a field is the smallest cardinal u such that every quadratic form in more than u variables over the field has a nontrivial zero. To determine this invariant for particular fields is in the focus of current quadratic form theory. Considering hermitian forms over a division algebra with involution gives rise to a corresponding notion of u-invariant. Here, we investigate this invariant for a quaternion algebra with involution. For several cases, we obtain upper and lower bounds on this invariant.

Keywords: hermitian form, isotropy, dimension, algebra with involu- tion, u-invariant

Classification (MSC 2000): 11E04, 11E39, 11E81

1 Summary

In this article we consider hermitian and skew-hermitian forms over a division algebra with involution over field of different from 2. The basics about hermitian forms over a division algebra with involution will be recalled in Section 2. In Section 3, we introduce the notion u-invariant of a division algebra with involution. For a division algebra of exponent two we define the orthogonal and the symplectic u-invariant. Section 4 shows how

∗Supported by the Swiss National Science Foundation (Grant No. 200020-100229/1) †Supported by the Irish Research Council for Science (Basic Research Grant SC/02/265)

1 Kneser’s Theorem, relating the classical u-invariant to the number of square classes of a given nonreal field, can be extended to our setting. In Section 5 we investigate the equivalence relation on pure quaternions induced by isom- etry of 1-dimensional skew-hermitian forms on a quaternion division algebra. We obtain an upper bound on the number of isometry classes of those forms. On the way to this result, we retrieve an exact sequence due to Lewis [10], which involves the set of 1-dimensional skew-hermitian forms over a quater- nion algebra and and the square class of a maximal subfield. Section 6 provides a fairly general method to prove the existence of 3-dimensional skew-hermitian forms over a quaternion division algebra with its canonical involution, showing that the orthogonal u-invariant can be 1 or 2 only in very special cases. In Section 7, we consider fields with a unique quaternion division algebra and obtain a complete classification of skew-hermitian forms in this case. These are straightforward generalisations classical results due to Jacobson [4] and Tsukamoto [14] concerning skew-hermitian forms over quaternion algebras over real closed fields and over local fields. In the final Section 8, we give general upper bounds for the hermitian u-invariants over a division algebra which is defined over a Ci-field, which is sharp for quaternion algebras.

2 Involutions and hermitian forms

Throughout this article K denotes a field of characteristic different from 2. Let D be a division whose center is equal to K and such that dimK (D) is finite; we refer to such D as a K-central division algebra, for short. We further assume that D is endowed with an involution σ, by which we mean a map σ : D → D such that σ ◦ σ is the identity on D and such that, for any a, b ∈ D, one has

σ(a + b) = σ(a) + σ(b) and σ(ab) = σ(b)σ(a)

Note that σ restricts to an involution of the center K. There are two cases to be distinguished. If σ|K is the identity on K, then we say that the involution σ is of the first kind. In the other case, when σ|K is a nontrivial automorphism of the field K, we say that σ is of the second kind. In general, we fix the subfield k = {x ∈ K | σ(x) = x} and say that σ is a K/k-involution of D.

2 Note that σ : D → D is k-linear. If σ is of the second kind, then K/k is a quadratic extension. Let ε ∈ K× with σ(ε)ε = 1. We are mostly interested in the cases where ε = ±1 and in fact, if σ is of the first kind then these are the only possibilities for ε. An ε-hermitian form over (D, σ) is –strictly speaking– a pair (V, h) where V is a finite-dimensional D-right module and h is a map h : V ×V → D which is K-linear in the second argument and which satisfies σ(h(x, y)) = ε · (h(y, x)) for any x, y ∈ V. It follows then that h is ‘sesquilinear’ in the sense that h(xa, yb) = σ(a) · h(x, y) · b for any x, y ∈ V , a, b ∈ D. We will, however, refer in this setting to h as the ε-hermitian form and to V as its underlying space. We simply say that h is hermitian (resp. skew- hermitian) if it is ε-hermitian for ε = 1 (resp. ε = −1). The simplest case occurs when D = K is a field, σ = idK , and ε = 1. In that case, a 1-hermitian form over K is a symmetric bilinear form b : V × V → K on a finite dimensional vector space V over K, and it can be identified (via the choice of a basis) to a quadratic form (i.e. a homogeneous polynomial of degree 2) over K in n = dimK (V ) variables. An ε-hermitian form (V, h) on (D, σ) is regular if, for any x ∈ V \{0}, the associated K-linear form V → K, y 7→ h(x, y) is nontrivial. We say that h is isotropic if there exists x ∈ V \{0} such that h(x, x) = 0. Let h1, h2 be two ε-hermitian forms over (D, σ), with underlying spaces V1,V2. The orthogonal sum h1 ⊥ h2 is the ε-hermitian form on the D-vector space V = V1 × V2 defined by h(x, y) = h1(x1, y1) + h2(x2, y2) for any x = (x1, x2), y = (y1, y2) ∈ V . An isometry between h1 and h2 is an of D-vector spaces τ : V1 → V2 such that h1(x, y) = h2(τ(x), τ(y)). If such τ exists, then h1 and h2 are isometric and we may write h1 ' h2 to indicate this; note that τ −1 then is an isometry, too. Witt’s Cancellation Theorem (cf. [7, (6.3.4) Theorem]) says that, whenever h1, h2 and h are ε-hermitian forms on (D, σ) such that h1 ⊥ h ' h2 ⊥ h, then one has an isometry h1 ' h2 as well. For a ε-hermitian form on (D, σ) we write ∆(h) = {h(x, x) | x ∈ V \{0}} ⊂ D. Note that this set contains 0 if and only if h is isotropic. We further denote Symε(D, σ) = {x ∈ D | σ(x) = εx} .

3 Elements of Symε(D, σ) are said to be ε-symmetric. For any ε-hermitian form ε h over (D, σ) we have ∆(h) ⊂ Sym (D, σ). Conversely, given a1, . . . , an ∈ Symε(D, σ), we define an ε-hermitian form h on the D-vector space V = Dn by h(x, y) = σ(x1)a1y1 + ··· + σ(xn)anyn n for any x = (x1, . . . , xn), y = (y1, . . . , yn) ∈ D = V . We denote this form h by ha1, . . . , ani and call this a diagonal form. Note that it is regular if and only if ai 6= 0 for i = 1, . . . , n. Since we assumed that char(K) 6= 2, any ε-hermitian form is isometric to some diagonal form (cf. [7, (6.2.4) Proposition]). ε We denote by Hermn(D, σ) the set of isometry classes of regular n- dimensional ε-hermitian forms over (D, σ). We have the surjection

ε ε Sym (D, σ) \{0} −→ Herm1(D, σ) , which maps a to the class of hai. Two elements a, b ∈ Symε(D, σ) are congruent if there exists c ∈ D such that a = σ(c)bc; we write a ∼σ b to indicate this. In fact, a and b are congruent if and only if the ε-hermitian forms hai and hbi over (D, σ) are isometric.

2.1 Remark. In the case where D = K and ε = 1, then the elements of ε Herm1(D, σ) are in natural one-to-one correspondence to the elements of the square class group K×/K×2, hence we may identify the two sets and thus 1 endow Herm1(D, σ) with a natural group structure. One can proceed in a similar way in the two (similarly simple) cases, first when D is a quaternion algebra and σ is the canonical involution on D, and second when σ is a unitary involution on a field D = K.

As in [14], we define the discriminant of a skew-hermitian form (V, h) over a quaternion division algebra D = (a, b)K as follows. Let (x1, ··· , xn) be a × × D-basis of V . Denoting by N the reduced norm from Mn(D) to K , we n ×2 put disc(h) = (−1) N(h(xi, xj)) mod K . For a more general definition of the discriminant of a hermitian form, see [1, Sect. 3].

2.2 Remark. Assume that D is a K-quaternion division algebra and let γ denote its canonical involution. For a ∈ K×, there exists a skew-hermitian form of dimension 1 and discriminant a over (D, γ) if and only if −a is represented by the pure part of the norm form of D. In particular, any skew-hermitian form of dimension 1 over (D, γ) has nontrivial discriminant.

4 Given an ε-hermitian forms h over (D, σ) and an element a ∈ k×, we define the scaled ε-hermitian form ah in the obvious way. Two ε-hermitian forms h, h0 over (D, σ) are said to be similar, if h0 ' ah for some a ∈ k×.

3 Hermitian u-invariants

We keep the setting of the previous section. According to [12, Ch. 9, Defini- tion 2.4], we define u(D, σ, ε) = sup {dim(h) | h anis. ε-herm. form over (D, σ)} ∈ N ∪ {∞} and call this the u-invariant of the triple (D, σ, ε). Assume now that the involution σ is of the first kind. Note that in this case D is of exponent at most 2, i.e. D ⊗K D is split. Moreover, the K- linearity of σ implies that ε = ±1. Involutions of the first kind are either of orthogonal or of symplectic type. It turns out (cf. [11]) that u(D, σ, ε) only depends on ε and on the type of σ. More precisely, given two involutions of first kind σ, τ on D one has u(D, σ, ε) = u(D, τ, ε) if σ and τ are of the same type and u(D, σ, ε) = u(D, τ, −ε) if they are of opposite type. For ε = ±1 we therefore set

u+(D) = u(D, σ, +1) and u−(D) = u(D, σ, −1) where we take for σ an arbitrary orthogonal involution on D, for the number obtained does not depend on the choice of σ. The number u+(D) is referred to as the orthogonal u-invariant and u−(D) as the symplectic u-invariant of D. It turns out by the previous that for any symplectic involution τ on D and ε = ±1 one has u(D, τ, ε) = u(D, σ, −ε). Let us now turn to the case of an involution of the second kind σ. Let k denote the subfield of K consisting of the elements fixed by σ. It turns out that u(D, σ, ε) depends only on k, in particular it does not depend on ε at all.

4 Kneser’s Theorem

We want to obtain upper bounds on the u-invariant of a division algebra with involution. In this section, we give such a bound in terms of the number

5 of one-dimensional (skew-)hermitian forms, under some condition in terms of levels of certain subalgebras. This will extend a famous observation by Kneser from the commutative case. Obviously, this is only of any use if the number of (isometry classes of) one-dimensional forms is finite. So the result obtained is applicable essentially when the center is a field with finitely many square classes. We recall the definition of the level and of the u-invariant of an algebra with involution. Let σ be an involution on a central simple K-algebra D. The level of σ is defined as

s(D, σ) = sup{m ∈ N | m × h1i is anisotropic over (D, σ)} ∈ N ∪ {∞}. Whenever s(D, σ) is finite, it is the smallest number m such that −1 can be written as a sum of m hermitian squares over (D, σ). 4.1 Theorem. Let D be a K-division algebra equipped with an involution σ. Let ε ∈ K be such that σ(ε)ε = 1. Let ψ be an ε-hermitian form over (D, σ) × and let α ∈ D be such that σ(α) = εα. Let CD(α) be the centralizer of K(α) in D. Suppose that s(CD(α), σ|CD(α)) < ∞. If ϕ = ψ ⊥ hαi is anisotropic then ∆(ψ) ( ∆(ϕ).

Proof: We write 0 = σ(d0)d0 + ··· + σ(ds)ds with s = s(CD(α), σ|CD(α)) and × d0, . . . , ds ∈ CD(α) . We suppose that ∆(ψ) = ∆(ϕ) and want to conclude that ϕ is isotropic. In fact we will show for any 0 ≤ i ≤ s, that the element α · (σ(d0)d0 + ··· + σ(di)di) belongs to ∆(ϕ); for i = s this means indeed that ϕ is isotropic. Our claim is obvious for i = 0, because α and thus ασ(d0)d0 are repre- sented by ϕ. Let now 1 ≤ i ≤ s and assume that the claim is established for i − 1. With α(σ(d0)d0 + ··· + σ(di−1)di−1) ∈ ∆(ϕ) = ∆(ψ), we obtain readily that α(σ(d0)d0 + ··· + σ(di−1)di−1) + ασ(di)di ∈ ∆(ϕ), finishing the argument. 

4.2 Corollary. Assume that s(CD(K(α)), σ) < ∞ holds for every α ∈ Symε(D, σ). Then ε u(D, σ, ε) 6 |Herm1(D, σ)| .

Proof: Let h ' ha1, . . . , ani be an anisotropic ε-hermitian form of dimension n over (D, σ). Let hi = ha1, . . . , aii for i = 1, . . . , n. By (4.1) we have ∆(h1) ( ∆(h2) ( ··· ( ∆(hn) = ∆(h). We conclude that h represents at

6 least n pairwise incongruent elements of Symε(D, σ). As a consequence we ε ε have |Herm1(D, σ)| > n and therefore |Herm1(D, σ)| > u(D, σ, ε). 

4.3 Remark. The hypothesis of the above corollary can be reformulated. In fact, this hypothesis is equivalent to having that s(CD(E), σ|CD(E)) < ∞ just for those (commutative) subfields E of D which are contained in Symε(D) and which are maximal with this property. We give a more explicit formulation of this for the case ε = 1:

(a) If σ is orthogonal, then the hypothesis means that every maximal sub- field E of D such that σ|E = idE is a nonreal field. (b) If σ is symplectic, then the hypothesis means that, for every subfield 1 E of D such that σE = idE and with [E : K] = 2 deg(D), the E- quaternion algebra CD(E) splits over every real closure of E. In order to see this, note that, if Q is an E-quaternion algebra, then the canonical involution of Q has finite level if and only if the norm form of Q is a torsion quadratic form over E, if and only if Q splits over every real closure of E. √ (c) If σ is unitary (i.e. of the second kind) and K = k( α) with α ∈ k, then the hypothesis means that α ∈ P E×2 for every extension E/k contained in D such that σE = idE and [E : k] = deg(D).

Moreover, the hypothesis of the√ corollary is trivially satisfied if k is nonreal or if σ is unitary and K = k( α) for some α ∈ P k×2. 4.4 Example. Let p be a different from 2 and let Q denote the unique quaternion division algebra over Qp. Then u+(Q) = |Herm−1(Q, γ)| = 3 .

Let now m be a positive integer and K = Qp((t1)) ... ((tm)). Then QK is a K-quaternion division algebra and we have

+ −1 m u (QK ) = |Herm (QK , γ)| = 3 · 2 .

This follows from the fact that the u-invariant(s) and the number of ε- hermitian forms over any division algebra defined over a field F each double when the algebra is extended to the power series field F ((t)).

7 5 Congruence of pure quaternions

In this section we consider a K-quaternion division algebra Q. We denote −1 the canonical involution of Q by γ. We want to describe Herm1 (Q, γ). Whenever a skew-hermitian form appears, it is meant to be skew-hermitian with respect to γ. 5.1 Proposition. Two one-dimensional skew-hermitian forms over (Q, γ) are similar if and only if their discriminants are the same (in K×/K×2). Proof: In more generality, any two skew-hermitian forms over (Q, γ) which are similar must have the same discriminant. Assume now that z1, z2 ∈ Q× are pure quaternions such that the discriminants of the skew-hermitian 2 forms hz1i and hz2i coincide. Observe that disc(hzii) = − Nrd(zi) = zi mod K×2, for i = 1, 2. Hence saying that the discriminants are the same × 2 2 2 2 means that there exists d ∈ K such that z2 = d z1 = (dz1) . Hence the × pure quaternions z2 and dz2 are congruent in Q, i.e. there exists α ∈ Q −1 such that dz1 = α z2α. Multiplying this equality by Nrd(α) = γ(α)α, we obtain Nrd(α)dz1 = γ(α)z2α . × We put c = Nrd(α)d ∈ K and obtain that cz1 ∼ z2. So, hz1i and hz2i are similar. 

5.2 Remark. A closer examination of the above argument yields the follow- ing refinement. Assume that G ⊂ K× is a subgroup containing Nrd(Q×). In order that two one-dimensional skew-hermitian forms are obtained one from each other by scaling with an element of G, it is necessary and sufficient to have that their discriminants coincide in K×/G2. The following is proved in [13, Chap. 10, 3.4. Lemma]. 5.3 Lemma (Scharlau). Let λ, µ ∈ Q× be anticommuting, so in particular 2 2 × × Q = (a, b)K for a = λ , b = µ ∈ K . Let c ∈ K . One has hλi ' hcλi as skew-hermitian forms over (Q, γ) if and only if c is represented over K by one of the quadratic forms h1, −ai and hb, −abi. In other words the lemma says that, for λ ∈ Symε(Q, σ)\{0} and c ∈ K×, one has hλi ' hcλi if and only if either c or bc is a norm of the quadratic extension K(λ)/K. The following exact sequence was obtained –in slightly different terms– by Lewis in [10].

8 −1 5.4 Proposition. We consider Herm1 (Q, γ) as a pointed set with distin- guished point given by some fixed pure quaternion λ ∈ Q×. With L = K(λ) and a = λ2 the sequence

× × · λ −1 −a Nrd(∗) × ×2 1 −→ Z/2Z −→ K /NL/K (L ) −→ Herm1 (Q, γ) −→ K /K is exact.

Proof: By (5.3) the group of elements x ∈ K× such that hxλi ' hλi is equal × × × to NL/K (L ) ∪ bNL/K (L ) where b ∈ K is such that Q = (a, b)K . This proves the exactness in the first two terms. We further have exactness at −1 Herm1 (Q, γ), by (5.1). 

5.5 Remark. J.-P. Tignol pointed out to us that the exact sequence of the previous proposition can also be obtained by a cohomological argument in the following way. Let ρ = Int(λ) ◦ γ. One has Herm1(Q, ρ) = H1(K,O(ρ)) where O(ρ) = {x ∈ Q : ρ(x)x = 1}. By [8, Chap. VII, §29], one has the exact sequence + 1 → O (ρ) → O(ρ) → µ2 → 1 , + 1 × where O (ρ) = L = {x ∈ L | NL/K (x) = 1} for L = K(λ). This yields an exact sequence

1 1 1 × ×2 1 → µ2 → H (K,L ) → H (K,O(ρ)) → K /K .

1 1 × × Finally, H (K,L ) ' K /NL/K (L ). 5.6 Proposition. Let π = h1i ⊥ π0 be the norm form of Q. Let

S = {aK×2 | a ∈ ∆(π0)} ⊂ K×/K×2 .

For a square class α ∈ S, set

−1 Hα = {h ∈ Herm1 (Q, γ) | disc(h) = α}. Then the following hold:

−1 S −1 P (1) Herm1 (Q, γ) = Hα , in particular |Herm1 (Q, γ)| = |Hα|. α∈S α∈S

1 × × √ (2) For any α ∈ S we have |Hα| 6 2 K /NL/K (L ) where L = K( α).

9 Proof: (1) is clear. For (2), we choose a pure quaternion λ such that disc(hλi) = α and apply the exact sequence in (5.4) with L = K(λ). 

−1 × ×2 5.7 Corollary. Herm1 (Q, γ) is finite if and only if K /K is finite. Proof: Assume that K×/K×2 is finite. Let S = {aK×2 | a ∈ ∆(π0)}, where π0 is the pure part of the norm form of Q. For any quadratic extension × × × ×2 L/K, the group K /NK(L/K (L ) is a quotient of K /K and thus finite, too. Using (2) of the last proposition, it follows that Hα is finite for any α ∈ S. Finally, S being a subset of K×/K×2 it is also finite. Hence (1) of −1 the proposition shows that Herm1 (Q, γ) is finite. −1 Conversely, assume that Herm1 (Q, γ) is finite. Let L be any maximal × × subfield of Q. Then K /NL/K (L ) is finite by the exact sequence. Since K×/ Nrd(Q×) is a quotient of this group, it is also finite. Moreover, the −1 × ×2 0 image of disc : Herm1 (Q, γ) −→ K /K is finite, and this means that π represents only finitely many square classes of K (i.e. S is finite). Since the group of reduced norms Nrd(Q×) is generated by the elements of K× which are represented by π0, it follows that Nrd(Q×)/K×2 is finite. We therefore × ×2 obtain that K /K is finite. 

5.8 Corollary. With the notation of the last proposition, we have

−1 1 × × Herm1 (Q, γ) 6 sup K /NL/K (L ) · |S| 2 L∈L where L denotes the set of maximal subfields of Q. 5.9 Remark. Kaplansky showed in [5] that Q is the unique quaternion division algebra over K if and only if

× × sup K /NL/K (L ) = 2 . L∈L

−1 In this case the corollary yields |Herm1 (Q, γ)| 6 |S|. Since the opposite in- −1 equality is obvious from the above proposition, we get |Herm1 (Q, γ)| = |S|. This applies in particular to any local field. Moreover, if K is a non- dyadic local field then we have |K×/K×2| = 4 and |S| = 3 and therefore + −1 obtain immediately that u (Q) = |Herm1 (Q, γ)| = |S| = 3. To illustrate this, let us consider Hamilton’s quaternion algebra Q = × ×2 (−1, −1)R. Since |R /R | = 2 and |S| = 1 for S as above, we conclude that −1 + |Herm1 (Q, γ)| = 1. Therefore u (Q) = 1.

10 6 Anisotropic forms of dimension three

We want to show for a large class of quaternion algebras that there exist 3- dimensional anisotropic hermitian forms w.r.t any orthogonal involution. In fact, to do this for any specific K-quaternion division algebra Q, it suffices to show the existence of a 3-dimensional skew-hermitian form over (Q, γ) where γ denotes the canonical involution.

6.1 Lemma. Let Q be a quaternion division algebra over K and let π denote its norm form and π0 the pure part of π. Let γ be the canonical involution of Q. Let h = hx, y, zi be a skew-hermitian form over (Q, γ). Suppose that Nrd(xyz) ∈/ ∆(π0). Then h is anisotropic.

Proof: If h were isotropic, then h ' H ⊥ hwi for some pure quaternion w; 0 then Nrd(xyz) = Nrd(w) ∈ ∆(π ), contradicting the hypothesis.  Recall that a preordering of a field K is a subgroup T ⊂ K× containing K×2 and such that T ∪ {0} is closed under addition. By the Artin-Schreier Theorem, the minimal preordering of K is P K×2, and a proper preordering T ( K× exists if and only if K is real (cf. [9, Chap. VIII, Sect. 9]). 6.2 Theorem. Let Q be a K-quaternion division algebra and let π = h1i ⊥ π0 denote its norm form. The following are equivalent:

(1) ∆(π0) is a preordering of K.

(2) ∆(π0) is closed under multiplication.

(3) ∆(π0) = ∆(π).

(4) For any a, b, c ∈ ∆(π0) one has abc ∈ ∆(π0).

Moreover, if any of these conditions is satisfied, then K is a real field and ∼ × Q = (−1, t)K for some t ∈ K . Proof: By the definition of a preordering, (1) implies (2). Since any element of Q is a product of two pure quaternions, the group of norms ∆(π) is gen- erated by the subset ∆(π0). Therefore (2) implies (3). Since ∆(π) is always a group, it is clear that (3) implies (4). Assume now that (4) holds. Take first an arbitrary diagonalisation π0 ' ha, b, ci. Since π0 has determinant 1, we have abc ∈ K×2. Further, by (4) and

11 since a, b, c ∈ ∆(π0), we obtain abc ∈ ∆(π0) and thus 1 ∈ ∆(π0). Fixing now c = 1 ∈ ∆(π0) we conclude from (4) that ∆(π0) is closed under multiplication. Hence, by the previous (2)–(4) are all satisfied. Let now a, b ∈ ∆(π0). Then a−1b ∈ ∆(π0) and thus 1 + a−1b ∈ ∆(π) = ∆(π0), by (3). Using (4) we obtain that a + b = a(1 + a−1b) ∈ ∆(π0). This shows that ∆(π0) is closed under addition. Together with (2) this yields that ∆(π0) is a preordering of K. Since ∆(π0) is itself closed under addition (i.e. without adding 0 to this set), it follows that K is a real field. Finally, we saw already that π0 represents 1, 0 ∼ × ∼ so π = h1, −t, −ti for some t ∈ K , whence Q = (−1, t)K . 

6.3 Corollary. Let Q be a K-quaternion division algebra and let π = h1i⊥π0 denote its norm form. If ∆(π) 6= ∆(π0), then u+(Q) ≥ 3. This is in partic- √ ular the case if K is nonreal or if QK( −1) is not split. Proof: Assume that ∆(π) 6= ∆(π0). By (6.2), there exist a, b, c ∈ ∆(π0) such that abc∈ / ∆(π0). Then there exist pure quaternions x, y, z ∈ Q× such that Nrd(x) = a, Nrd(y) = b, and Nrd(z) = c. We consider the skew-hermitian form h = hx, y, zi over (Q, γ). By (6.1), h is anisotropic. + √ Therefore u (Q) ≥ 3. Furthermore, if K is nonreal or if QK( −1) is not split, 0 then ∆(π) 6= ∆(π ) by (6.2). 

7 Kaplansky fields and hermitian forms

Kaplansky studied first the condition on a field K that there exists a unique 3-dimensional anisotropic quadratic form of trivial determinant over K (cf. [5]). Fields with this property have been referred to as ‘pre-Hilbert fields’ (cf. [9, p. 453]). However, since they are a post-Hilbert invention, due to Kaplansky, we prefer to use the term ‘Kaplansky field’ instead. Note that there are many (equivalent) characterisations of Kaplansky fields. In particular, K is a Kaplansky field if and only if there exists (up to isomorphism) a unique nonsplit K-quaternion algebra.

7.1 Lemma. Let K be a Kaplansky field. Let Q denote the unique K- quaternion division algebra and γ its canonical involution. For any pure quaternion λ ∈ Q× and any d ∈ K× we have hλi ' hdλi as skew-hermitian forms over (Q, γ).

12 Proof: Let µ ∈ Q× be such that µλ = −λµ. In particular we have Q ' 2 2 (a, b)K for a = λ and b = µ . Assume that hλi 6' hdλi. According to (5.3), then d is represented neither by h1, −ai nor by hb, −abi. Then (a, d)K is a quaternion algebra which is neither split nor isomorphic to (a, b)K , that is, to Q. This contradicts the hypothesis. 

7.2 Theorem. Let K be a Kaplansky field. Let Q be the unique quaternion division algebra over K and γ its canonical involution.

(1) Skew-hermitian forms over (Q, γ) are classified by their dimension and discriminant.

(2) A two-dimensional skew-hermitian form over (Q, γ) is isotropic if and only if it has trivial discriminant.

(3) Any three-dimensional skew-hermitian form over (Q, γ) with trivial dis- criminant is anisotropic.

(4) Every skew-hermitian form of dimension > 3 over (Q, γ) is isotropic.

− Proof: We first show (1) for 1-dimensional forms. Let z1, z2 ∈ Sym (Q, γ) and assume that the skew-hermitian forms hz1i and hz2i over (Q, γ) have the same discriminant. According to (5.1), then hz1i ' hcz2i for some c ∈ K. Since also hz2i ' hcz2i by (7.1), we obtain that hz1i ' hz2i. This shows that 1-dimensional skew-hermitian forms over (Q, γ) are classified by their discriminant. − (2) Let z1, z2 ∈ Sym (Q, γ) be such that the form hz1, z2i has discrim- × ×2 inant 1. Then Nrd(z1) and Nrd(z2) represent the same class in K /K . This means that the 1-dimensional forms hz1i and h−z2i have the same dis- criminant, whence hz1i ' h−z2i by what we showed above. (3) Let ϕ be a form of dimension 3 with trivial discriminant. If ϕ is isotropic then we have a decomposition ϕ ' H⊥hai. The discriminant of hai becomes trivial. Contradiction to what we stated in (2). (4) We give Tsukamoto’s argument. Let ϕ be a 4-dimensional skew- hermitian form over (Q, γ). We may assume that ϕ = ha1, a2, a3, a4i, with − − a1, . . . , a4 ∈ Sym (Q, γ). Since Sym (Q, γ) has K-dimension 3, there exist c1, . . . , c4 ∈ K, not all zero, such that

c1a1 + c2a2 + c3a3 + c4a4 = 0 .

13 It follows from what we showed at the beginning of the proof, that there exist d1, . . . , d4 ∈ Q such that ciai = γ(di)aidi for i = 1,..., 4. We obtain P4 then i=1 γ(di)aidi = 0. Therefore ϕ is isotropic. (1) Let ϕ and ψ be two n-dimensional skew-hermitian forms over (Q, γ), where n ≥ 1 is arbitrary, and assume that both forms have the same discrim- inant. By (3), the 2n-dimensional form ϕ ⊥ −ψ then splits off at least n − 1 hyperbolic planes. The remaining 2-dimensional form has discriminant 1 and thus is hyperbolic by (2). Therefore ϕ ⊥ −ψ is hyperbolic, which means that ϕ ' ψ.  Actually, both conditions (1) and (2) in the above theorem are equivalent to the assumption made, that Q is the unique K-quaternion division algebra. 7.3 Corollary. Let Q be a K-quaternion division algebra and γ its canonical involution. Skew-hermitian forms over (Q, γ) are classified by dimension and discriminant if and only if Q is the unique K-quaternion division algebra. Proof: It is clear from the theorem, that the condition is sufficient. In order to show that it is necessary, assume that Q is not the unique K-quaternion division algebra. Then there exists some λ ∈ Q \ K such that the quadratic × extension L = K(λ), contained in Q, is such that NL/K (L ) has index at × × 2 least 4 in K . Let a, b ∈ K such that λ = a and Q ' (a, b)K . By the previous, there exists some element c ∈ K× such that neither c nor bc is a norm of L/K. Then the two 1-dimensional skew-hermitian forms hλi and hcλi over (Q, σ) have the same discriminant, but they are not isometric, by (5.3).  From the theorem we obtain a generalisation of a result due to Tsukamoto on quaternion-division algebras over local fields. 7.4 Corollary. Let K be a nonreal Kaplansky field and let Q be the unique quaternion division algebra over K. Then u+(Q) = 3. Proof: By the theorem, u+(Q) ≤ 3. The opposite inequality was shown in (6.3).  A field K is said to be euclidean if the squares in K form an ordering. Equivalently, K is euclidean if and only if it is real and such that K× = K×2 ∪ −K×2. (cf. [9, Chap. VIII, Proposition 4.2.]). If K is euclidean, then (−1, −1)K is the unique K-quaternion division algebra, so that K is in particular a Kaplansky field.

14 7.5 Proposition. Let Q be a K-quaternion division algebra with the canon- ical involution γ. Let π denote the norm of Q and π0 the pure part π. The following are equivalent:

(1) u+(Q) = 1.

−1 (2) |Herm1 (Q, γ)| = 1.

(3) K is euclidean and Q ' (−1, −1)K . Moreover, when these equivalent conditions hold we have |∆(π0)/K×2| = 1 and Nrd(Q×) = K×2.

Proof: The equivalence of (1) and (2) is clear. Suppose now that (1) and (2) are satisfied. We have to show that K is euclidean and Q ' (−1, −1)K . −1 0 ×2 0 Since |Herm1 (Q, γ)| = 1 we have ∆(π ) = K . It follows that π ' h1, 1, 1i, P ×2 ×2 whence Q ' (−1, −1)K . Furthermore K = K , i.e. K is pythagorean. If K were nonreal, then −1 ∈ P K×2 = K×2 would yield that Q is split, which not the case. Hence K is real. It remains to show that K is euclidean, i.e. that K× = −K×2 ∪ K×2. Let a ∈ K×. We fix i ∈ Q such that i2 = −1 and L = K(i). By (1), the hermitian form h = hi, aii is isotropic. × × ×2 ×2 Therefore −a ∈ (NL/K (L ) ∪ −NL/K (L )) = −K ∪ K by (5.3), whence a ∈ −K×2 ∪ K×2. Assume now that (3) holds. Then K is a Kaplansky field. Since all 1- dimensional skew-hermitian forms over (Q, γ) have trivial discriminant, (7.2) says that they are all isometric, showing (2).  Let us now give an example of a real Kaplansky field which is not eu- clidean and where we have a more direct argument why u+((−1, −1)) > 1 in this case.

7.6 Example. Let K be an extension of Q in R which is maximal with respect to the property that 2 is not a square in K. Then K is a real field with four square classes represented by ±1, ±2. The unique quaternion division algebra over K is Q = (−1, −1)K . However, since 2 is a sum of two squares in K, we can write Q = (−1, −2)K and thus there exists anticommuting α, β ∈ Q with α2 = 1, β2 = 2. Then the skew-hermitian form h = hα, βi over (Q, γ) (with γ the canonical involution on Q) has nontrivial discriminant disc(h) = 2 and thus is anisotropic. This shows that u+(Q) = 2.

15 7.7 Proposition. Let K be a real Kaplansky field and let Q = (−1, −1)K . Then u+(Q) ≤ 2.

Proof: Let γ be the canonical involution on Q and let i be a pure quaternion in Q with i2 = −1. By (7.2), the form hi, ii is hyperbolic. We claim that every two-dimensional skew-hermitian form over (Q, γ) is of the shape h ∼= hi, zi for some pure quaternion z ∈ Q×. Once this claim is established it follows that every three-dimensional skew-hermitian form over (Q, γ) contains hi, ii and therefore is isotropic. So, let h be a two-dimensional skew-hermitian form over (Q, γ). Let a ∈ K× represent its discriminant. Then a ∈ Nrd(Q×). The reduced norms of Q are elements of K× which are sums of (four) squares in K. Since K is a real Kaplansky field, we have p(K) ≤ 2. (In fact, for any P ×2 a ∈ K , the quaternion algebra (−1, a)K is split, for it cannot be isomor- × phic to (−1, −1)K .) In particular every reduced norm of Q is actually the reduced norm of some pure quaternion. Let z be a pure quaternion in Q such that Nrd(z) = a. Then the skew-hermitian form hi, zi has also discriminant a and is therefore isometric to h, by (7.2). 

7.8 Example. Let m be a nonnegative integer. Then there exists a real Kaplansky field K such that |K×/K×2| = 2m. In fact, let W denote the product in the category of abstract Witt rings of W (R) = Z with

W (C((X1)) ... ((Xm−1))) = Z[G] where G = (Z/2Z)m−1. By the results in [6], there exists a field K such that W (K) = W , and it follows that K has the desired properties. Now, if m ≥ 1 + we obtain that u ((−1, −1)K ) = 2. 7.9 Corollary. Let K be a Kaplansky field and let Q be the unique quaternion division algebra over Q. Then we have   1 if K is real euclidean, u+(Q) = 2 if K is real non-euclidean,  3 if K is nonreal.

16 8 Hermitian u-invariant and Ci-fields

A field K is called a Ci-field if every homogeneous polynomial over K of degree d in at least di + 1 variables has a non-trivial zero. We recall the most prominent (known) examples of such fields. If F/C is an extension of transcendence degree i where C is algebraically closed, then F is a Ci-field. Similarly, if F/F is an extension of transcendence degree i − 1 (i ≥ 1) of a finite field F, then F is a Ci-field. For an introduction to the theory of Ci- fields we recommend [3]. For more recent results and open problems in this context, the reader is referred to [12], where also variations of the Ci-property are discussed. For a field K and a r, one denotes by ur(K) the supremum over the natural numbers n such that there exists a system of r quadratic forms in the variables X1,...,Xn over K which have no common zero other than the trivial one. The numbers ur(K) are also called the system u- invariants of K. In particular one has u1(K) = u(K). It is well known that, if K is a Ci-field, then

i ur(K) 6 r · 2 for any r ≥ 1 . Using the following result from [11, Proposition 3.6], we will obtain esti- mates for the u-invariants of division algebras with involution over a Ci-field. 8.1 Proposition. Let D be a division algebra of degree m over its center K with a K/k-involution σ and let ε ∈ K with εσ(ε) = 1. Then u (k) u(D, σ, ε) r 6 m2[K : k] where r is the dimension of k-vector space Symε(D, σ). The proof of this proposition is based on the fact that to every ε-hermitian form h over (D, σ) one can associate a system of r symmetric bilinear forms over k in such a way that the isotropy of h is equivalent to the simultaneous isotropy of this system.

8.2 Corollary. Let K be a Ci-field and let D be a K-division algebra of + i−1 m+1 − i−1 m−1 exponent 2 and of degree m. Then u (D) 6 2 · m and u (D) 6 2 · m . i Proof: We use the previous proposition and the fact that ur(k) 6 2 r. 

17 8.3 Corollary. Let K be a Ci-field. Let D be a K-quaternion division algebra. Then + i−2 u (D) 6 3 · 2 . The bound given by the corollary is sharp for any i, as one can see from the most natural examples.

8.4 Example. Let k be a C2-field (e.g. k = C(X,Y )) and let Q be a k- + quaternion division algebra (e.g. Q = (X,Y )k). Then u (Q) ≥ 3 by (6.3). On the other hand u+(Q) ≤ 3 by the last corollary. Therefore u+(Q) = 3. Let now K = k((t1)) ... ((tn−2)) for some integer n > 2. Then K is a Cn-field and + n−2 u (QK ) = 3 · 2 . 0 Moreover, this equality still holds if K is replaced by K = k(t1, . . . , tn−2), which is also a Cn-field.

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Karim Johannes Becher, Ecole´ Polytechnique F´ed´erale de Lausanne, SB IMB CSAG, MA C3 595, Station 8, CH - 1015 Lausanne, Switzerland. Email: [email protected]

Mohammad Golamzadeh Mahmoudi, Department of Mathematics, University College Dublin, Belfield, Dublin 4, Ireland. Email: [email protected]

31 May 2005

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