DECOMPOSABLE QUADRATIC FORMS AND INVOLUTIONS

D.W. LEWIS, M.G. MAHMOUDI, J.-P. TIGNOL, AND O. VILLA

Abstract. In his book on compositions of quadratic forms, Shapiro asks whether a quadratic form decomposes as a tensor product of quadratic forms when its adjoint involution decomposes as a tensor product of involutions on central simple algebras. We give a positive answer for quadratic forms defined over local or global fields and produce counterexamples over fields of rational fractions in two variables over any formally real field.

1. Introduction

Every nondegenerate symmetric or skew-symmetric bilinear form b on a finitedimensional vector space V induces an involution (i.e. an anti-automorphism of period 2) on the endomorphism algebra End V. This involution, known as the adjoint involution of b and denoted by ad_{b}, is characterized by the following property:

b(x, f(y)) = b(ad_{b}(f)(x), y) for x, y ∈ V and f ∈ End V .

If the form b is the tensor product of two nondegenerate symmetric or skewsymmetric bilinear forms, i.e.

V = V_{1} ⊗ V_{2}  and  b = b_{1} ⊗ b_{2},

then it is easy to see that the adjoint involution also decomposes,

End V = (End V_{1}) ⊗ (End V_{2})  and  ad_{b} = ad_{b_{1}} ⊗ ad_{b_{2}},

(see [12, Corollary 6.10]), which we denote

(End V, ad_{b}) = (End V_{1} ⊗ End V_{2}, ad_{b_{1}} ⊗ ad_{b_{2}}).

In [12, p. 201], Shapiro raises the following question: if the adjoint involution has a nontrivial decomposition of the form

(End V, ad_{b}) ≃ (A_{1}, σ_{1}) ⊗ (A_{2}, σ_{2})

for certain involutions σ_{1}, σ_{2} on central simple algebras A_{1}, A_{2}, does it follow that b decomposes nontrivially into a tensor product of bilinear forms? He showed that the answer is positive if A_{1} and A_{2} are split, see [12, Corollary 6.10] or Proposition 3.1 below.

In this paper, we give an affirmative answer to Shapiro’s question over certain fields of characteristic different from 2, including local and global fields, and give examples where the answer is negative. Our examples are involution trace forms of central simple algebras of degree 2m, where m is an odd integer, see Subsection 3.2.

Date: 21 February 2005.

The second and third authors were partially funded by IRCSET Basic Research Grant SC/02/265. The third author was partially supported by the National Fund for Scientific Research (Belgium). He gratefully acknowledges the hospitality of University College Dublin while this paper was written. The fourth author acknowledges the financial support provided through the European Community’s Human Potential Programme under contract HPRN-CT-2002-00287, KTAGS.

Throughout the paper, the characteristic of the base field F is assumed to be different from 2. Therefore, symmetric bilinear forms correspond bijectively to quadratic forms, and this correspondence is used to define the tensor product of quadratic forms. For any (nondegenerate) quadratic form q, we write ad_{q} for the adjoint involution of the polar form of q.

In Section 2, we collect results on factorizations of quadratic forms. In particular, we determine the decomposable forms of dimension at most 8. The decomposition of involutions is discussed in Section 3.

2. Decomposable quadratic forms

All the quadratic forms considered in this paper are nondegenerate. A quadratic form q over a field F is called decomposable if there is a factorization q ≃ q_{1}q_{2} where q_{1} and q_{2} are quadratic forms with dim q_{1}, dim q_{2} ≥ 2. In particular, the hyperbolic forms of dimension n ≥ 4 are decomposable since they are isometric to

⟨1, −1⟩ ⊗ (\frac{n}{2}⟨1⟩).

From the outset, note that the notion of decomposable form is not compatible with Witt equivalence, since every quadratic form q is Witt-equivalent to the decomposable form q ⊗ ⟨1, 1, −1⟩.

After reviewing below some general properties of factorizations of quadratic forms, we consider factorizations of forms of small dimension, and determine the indecomposable forms over some special fields.

2.1. General properties.

Lemma 2.1. If q_{1}, q_{2} are even-dimensional quadratic forms, the discriminant and Clifford invariant of the product q_{1}q_{2} satisfy

disc(q_{1}q_{2}) = 1  and  c(q_{1}q_{2}) = (disc q_{1}, disc q_{2})_{F} .

Proof. For i = 1, 2, let d_{i} ∈ F^{×} be such that disc q_{i} = d_{i}F^{×2}. Since q_{i} is evendimensional, we have q_{i} ≡ ⟨1, −d_{i}⟩ mod I^{2}F, hence

q_{1}q_{2} ≡ ⟨1 − d_{1}⟩⟨1, −d_{2}⟩ mod I^{3}F.

Therefore,

disc(q_{1}q_{2}) = disc(⟨1, −d_{1}⟩⟨1, −d_{2}⟩) = 1

and

c(q_{1}q_{2}) = c(⟨1, −d_{1}⟩⟨1, −d_{2}⟩) = (d_{1}, d_{2})_{F} .
□

For any quadratic form q over F, let q_{a} denote the anisotropic kernel of q, and let

dim_{a} q = dim q_{a}.

Lemma 2.2. If a quadratic form q satisfies dim q = n_{1}n_{2} for some integers n_{1}, n_{2} subject to

(1)  n_{1} ≡ dim_{a} q mod 2,  n_{2} ≡ 1 mod 2,

n_{1} ≥ dim_{a} q, n_{1}, n_{2} ≥ 2, (2)

then q is decomposable.

Proof. Letting k_{1} = \frac{1}{2}(n_{1} − dim_{a} q) and k_{2} = \frac{1}{2}(n_{2} − 1), we have

q = (q_{a} ⊥ k_{1}⟨1, −1⟩) ⊗ (⟨1⟩ ⊥ k_{2}⟨1, −1⟩).
□

Besides these (essentially trivial) factorizations, a main source of decompositions is the following well-known result (due to Pfister):

Lemma 2.3. An anisotropic quadratic form q splits over a quadratic extension F (\sqrt{d}) (i.e., becomes hyperbolic over F (\sqrt{d})) if and only if q ≃ ⟨1, −d⟩ ⊗ q′ for some quadratic form q′.

Proof. See [11, Chapter 2, Theorem 5.2]. □

Corollary 2.4. Let q be a decomposable quadratic form of dimension dim q = 2p for some odd prime number p. If disc q = 1, then q is hyperbolic. If disc q ≠ 1, then dim_{a} q ≡ 2 mod 4 and q splits over F (\sqrt{disc q}).

Proof. Let q = q_{1}q_{2} be a nontrivial decomposition of q. We may assume dim q_{1} = 2 and dim q_{2} = p. Comparing discriminants, we obtain disc q = disc q_{1} since p is odd. Therefore, if disc q = 1, then q_{1} is hyperbolic, hence q also is hyperbolic. If disc q = dF^{×2} ≠ 1, then F (\sqrt{d}) splits q_{1}, hence also q and q_{a}. By Lemma 2.3, it follows that q_{a} ≃ ⟨1, −d⟩ ⊗ q′ for some quadratic form q′. If dim q′ is even, then disc q_{a} = 1, a contradiction since disc q = disc q_{a}. Therefore, dim_{a} q ≡ 2 mod 4. □

The results above yield necessary and sufficient conditions for the decomposability of forms q with dim_{a} q ≤ 4, as we now show.

Proposition 2.5. A quadratic form q of odd dimension is decomposable whenever its dimension has a nontrivial factorization dim q = n_{1}n_{2} with n_{1} ≥ dim_{a} q. In particular, if dim_{a} q ≤ 3, then q is decomposable if and only if dim q is neither 1 nor a prime number.

Proof. If dim q is odd, then dim_{a} q is odd and the conditions (1) hold for any factorization dim q = n_{1}n_{2}. The proposition follows from Lemma 2.2. □

Proposition 2.6. If dim_{a} q = 2, then q is indecomposable if and only if dim q is a power of 2.

Proof. Note that dim q ≡ dim_{a} q mod 2, so the hypothesis implies dim q is even. If it has an odd prime factor p, let n_{1} = (dim q)/p and n_{2} = p. Then n_{1} is even, so q is decomposable by Lemma 2.2. Therefore, q is decomposable if dim q is not a power of 2.

Conversely, if dim q is a power of 2, then every nontrivial factorization of q has the form q = q_{1}q_{2} with dim q_{1}, dim q_{2} even. Then disc q = 1 by Lemma 2.1, which is impossible since dim_{a} q = 2. □

Proposition 2.7. If dim_{a} q = 4, then q is indecomposable if and only if one of the following conditions holds:

(a) dim q = 2p for some odd prime p,

(b) dim q is a power of 2 and disc q ≠ 1.

Proof. Since dim_{a} q = 4, it follows that dim q is even. In case (a), q is indecomposable by Corollary 2.4. If dim q is a power of 2, then every nontrivial factorization of q is into a product of forms of even dimension. The existence of such a factorization implies disc q = 1 by Lemma 2.1, hence q is indecomposable in case (b). Conversely, Lemma 2.2 shows that q is decomposable whenever dim q has a factorization dim q = n_{1}n_{2} with n_{1} even, n_{2} odd and n_{1} ≥ 4, n_{2} ≥ 3. Since dim q is even, such a factorization exists unless dim q is a power of 2 or of the form 2p for some odd prime p. To prove that either (a) or (b) holds when q is indecomposable, it only remains to show that q is decomposable if dim q is a power of 2 and disc q = 1. In this case, q_{a} has a diagonalization

q_{a} = ⟨a, b, c, d⟩

with abcd ∈ F^{×2}, so

q_{a} = ⟨a, b⟩ ⊗ ⟨1, ac⟩

and, for k = \frac{1}{4}(dim q) − 1,

q = ⟨a, b⟩ ⊗ (⟨1, ac⟩ ⊥ k⟨1, −1⟩).
□

2.2. Forms of small dimension. We may also use Propositions 2.6 and 2.7, together with Corollary 2.4, to discuss the decomposability of forms of dimension at most 8. Of course, quadratic forms whose dimension is 1 or a prime number are indecomposable. Therefore, we need to consider only forms of dimension 4, 6 or 8.

Proposition 2.8. A quadratic form q of dimension 4 is indecomposable if and only if one of the following conditions holds:

(a) dim_{a} q = 2,

(b) dim_{a} q = 4 and disc q ≠ 1.

Proof. This readily follows from Propositions 2.6 and 2.7. □

Proposition 2.9. A quadratic form q of dimension 6 is indecomposable if and only if one of the following conditions holds:

(a) dim_{a} q = 4,

(b) dim_{a} q = 6 and disc q = 1,
(c) dim_{a} q = 6, disc q ≠ 1 and q is not split by F (\sqrt{disc q}).

Proof. The form q is decomposable if dim_{a} q = 2, by Proposition 2.6, and indecomposable if dim_{a} q = 4, by Proposition 2.7. Therefore, it only remains to consider the case where dim_{a} q = 6, i.e., q is anisotropic. Corollary 2.4 shows that q is indecomposable in cases (b) and (c), and Lemma 2.3 proves that q is decomposable if disc q ≠ 1 and q is split by F (\sqrt{disc q}). □

Proposition 2.10. A quadratic form q of dimension 8 is indecomposable if and only if one of the following conditions holds:

(a) dim_{a} q = 2,

(b) dim_{a} q = 4 and disc q ≠ 1,

(c) dim_{a} q = 6,

(d) dim_{a} q = 8 and q is not split by any quadratic extension of F.

Proof. If dim_{a} q = 2, then q is indecomposable by Proposition 2.6. If dim_{a} q = 4, then q is indecomposable if and only if disc q ≠ 1, by Proposition 2.7. Suppose next dim_{a} q = 6. If disc q ≠ 1, then q is not decomposable by Lemma 2.1. If disc q = 1, then q is Witt-equivalent to an anisotropic Albert form, hence its Clifford invariant has index 4, by [5]. On the other hand, forms which decompose into a product of forms of even dimension have a Clifford invariant of index at most 2 by Lemma 2.1; therefore q is indecomposable. Finally, the case where dim_{a} q = 8 follows from Lemma 2.3. □

2.3. Forms over special fields. Recall from [11, Chapter 2,§16] that the uinvariant of a field F is the supremum of the dimensions of anisotropic quadratic forms over F.

Corollary 2.11. If u(F) ≤ 4, a quadratic form q over F is indecomposable if and only if one of the following conditions holds:

(a) dim q is a prime number,

(b) dim_{a} q = 2 and dim q is a power of 2,

(c) dim_{a} q = 4 and dim q = 2p for some odd prime p,

(d) dim_{a} q = 4, dim q is a power of 2 and disc q ≠ 1. In particular, every quadratic form q with dim q ≡ 0 mod 4 and disc q = 1 is decomposable.

Proof. The hypothesis u(F) ≤ 4 implies dim_{a} q ≤ 4. The corollary therefore readily follows from Propositions 2.5, 2.6 and 2.7. For the last statement, observe that dim_{a} q ≠ 2 if disc q = 1. □

The corollary applies for instance to p-adic fields or to non-formally real number fields. Note that case (d) does not arise for p-adic fields since every anisotropic quadratic form of dimension 4 then has trivial discriminant.

Suppose now F is a real-closed field. Quadratic forms over F are classified by their dimension and signature, which are related by

| sgn q| ≤ dim q  and  sgn q ≡ dim q mod 2.

Proposition 2.12. A quadratic form q over a real-closed field F is decomposable if and only if dim q and sgn q have factorizations of the form

dim q = n_{1}n_{2}, sgn q = s_{1}s_{2}

with

n_{i} ≥ 2, |s_{i}| ≤ n_{i} and s_{i} ≡ n_{i} mod 2 for i = 1, 2.

Proof. If q = q_{1}q_{2} is a nontrivial decomposition, then

dim q = dim q_{1} dim q_{2}  and  sgn q = sgn q_{1} sgn q_{2}

are factorizations of the required form. Conversely, given factorizations as in the statement of the proposition, one may find quadratic forms q_{1}, q_{2} over F with dim q_{i} = n_{i} and sgn q_{i} = s_{i}. The forms q and q_{1}q_{2} have the same dimension and signature, hence q = q_{1}q_{2}. □

Corollary 2.13. If a quadratic form q over a real-closed field F satisfies

dim q ≡ 0 mod 4  and  disc q = 1,

then q = ⟨1, 1⟩ ⊗ q′ for some quadratic form q′. In particular, q is decomposable.

Proof. Since dim q is even and disc q = 1, it follows that sgn q ≡ 0 mod 4. Therefore, we may apply the proposition above with n_{1} = s_{1} = 2. □

The case of quadratic forms of dimension divisible by 4 and trivial discriminant is particularly relevant for Shapiro’s question, see Section 3. The following result (of which Corollary 2.13 is a special case) gives further examples where these forms have a nontrivial decomposition.

Proposition 2.14. Suppose the field F satisfies the following properties:

• F is linked, i.e., every tensor product of quaternion F -algebras is Brauerequivalent to a quaternion algebra,

• every quaternion F -algebra is split by a quadratic extension K/F such that

I^{3}K = 0. Then every quadratic form q over F such that dim q ≡ 0 mod 4 and disc q = 1 decomposes as q = ⟨1, a⟩ ⊗ q′ for some a ∈ F^{×} and some quadratic form q′. If F is formally real, the element a may be chosen to be totally positive, i.e., a >_{P} 0 for every ordering P of F.

Proof. Since F is linked, the Clifford invariant c(q) is Brauer-equivalent to a quaternion F -algebra. Let K/F be a quadratic extension with I^{3}K = 0 which splits c(q). As was shown by Elman and Lam [3, Theorem 3.11], quadratic forms over K are classified by their dimension, discriminant and Clifford invariant, hence q splits over K. If K = F (\sqrt{d}), Lemma 2.3 yields

q_{a} = ⟨1, −d⟩ ⊗ q_{0}

for some quadratic form q_{0}. Since disc q = disc q_{a} = 1, it follows that dim q_{0} is even, hence dim q_{a} ≡ 0 mod 4. Therefore, the Witt index i_{W} (q) is even. Letting i_{W} (q) = 2m, we have

q = q_{a} ⊥ 2m⟨1, −1⟩ = ⟨1, −d⟩ ⊗ (q_{0} ⊥ m⟨1, −1⟩).

We obtain the required factorization with a = −d and q′ = q_{0} ⊥ m⟨1, −1⟩. Note that the condition I^{3}K = 0 implies K is not formally real, hence d <_{P} 0 for every ordering P of F (if any). □

Recall that the virtual cohomological 2-dimension of a field F is the cohomological 2-dimension of F (\sqrt{−1}). Fields with virtual cohomological 2-dimension at most 1 satisfy the conditions in Proposition 2.14 since I^{2}F (\sqrt{−1}) = 0.

To obtain further examples, we use the Effective Diagonalization Property (ED), which was characterized in various ways by Prestel and Ware [9]. For instance, a field F is ED if and only if for every valuation v with formally real residue field \overline{F}, the group 2v(F^{×}) has index at most 2 in v(F^{×}), and \overline{F} is euclidean when 2v(F^{×}) ≠ v(F^{×}). This readily shows that number fields are ED.

Lemma 2.15. Every quaternion division algebra over an ED field F is split by a non-formally real quadratic extension of F.

Proof. Let Q = (a, b)_{F} be a quaternion division F -algebra. Since F is ED, Theorem 2 of [9] shows that the form ⟨1, a, b, −ab⟩ is almost isotropic, which means that for every diagonalization ⟨1, a, b, −ab⟩ ≃ ⟨c_{1}, c_{2}, c_{3}, c_{4}⟩, there is an integer n such that n⟨c_{1}⟩ ⊥ ⟨c_{2}, c_{3}, c_{4}⟩ is isotropic. In particular, there is an integer n such that n⟨1⟩ ⊥ ⟨a, b, −ab⟩ is isotropic, hence ⟨a, b, −ab⟩ represents an element u ∈ F^{×} such that −u is a sum of squares. The algebra Q is then split by F (\sqrt{u}), which is a non-formally real quadratic extension of F. □

Corollary 2.16. If a quadratic form q over a global field F satisfies

dim q ≡ 0 mod 4  and  disc q = 1,

then q = ⟨1, a⟩ ⊗ q′ for some totally positive a ∈ F^{×} and some quadratic form q′. In particular, q is decomposable.

Proof. Global fields are linked and ED, and non-formally real global fields K satisfy I^{3}K = 0. Therefore, the corollary follows from Proposition 2.14 and Lemma 2.15.

□

3. Decomposable adjoint involutions

Let q be a quadratic form on an F -vector space V and let ad_{q} be the adjoint involution on End_{F} V. In this section, we consider decompositions of algebras with involution

(3)  (End_{F} V, ad_{q}) ≃ (A_{1}, σ_{1}) ⊗_{F} (A_{2}, σ_{2})

where A_{1}, A_{2} are central simple F -algebras and σ_{1}, σ_{2} are involutions on A_{1} and A_{2} respectively. Since End_{F} V represents the trivial element in the Brauer group of F, the Brauer classes of A_{1} and A_{2} are opposite. However, σ_{1} yields an isomorphism between A_{1} and its opposite algebra, hence A_{1} and A_{2} are also Brauer-equivalent. Since ad_{q} is an orthogonal involution, the involutions σ_{1} and σ_{2} must be both orthogonal or both symplectic, by [12, Proposition 6.9] or [5, (2.23)].

In [12, p. 201], Shapiro asks whether a nontrivial decomposition as in (3) implies that q is decomposable.

3.1. Positive results on Shapiro’s question. The case where A_{1} and A_{2} are split and σ_{1}, σ_{2} are orthogonal is discussed in Shapiro’s book [12]:

Proposition 3.1. Let q, q_{1}, q_{2} be quadratic forms on F -vector spaces V, V_{1}, V_{2}. The following statements are equivalent:

(a) (End_{F} V, ad_{q}) ≃ (End_{F} V_{1}, ad_{q_{1}}) ⊗_{F} (End_{F} V_{2}, ad_{q_{2}});

(b) there exists λ ∈ F^{×} such that q ≃ ⟨λ⟩q_{1}q_{2}.

Proof. See [12, Corollary 6.10, p. 112]. □

Corollary 3.2. Suppose that one of the algebras A_{1}, A_{2} in (3) is split. Then A_{1} and A_{2} are both split and the quadratic form q is decomposable if deg A_{1}, deg A_{2} > 1. Moreover, if σ_{1} and σ_{2} are symplectic, then q is hyperbolic.

Proof. The first part is clear, since A_{1} and A_{2} are Brauer-equivalent. Let A_{1} = End_{F} V_{1} and A_{2} = End_{F} V_{2} for some F -vector spaces V_{1}, V_{2}. If σ_{1} and σ_{2} are orthogonal, then there exist quadratic forms q_{1} on V_{1} and q_{2} on V_{2} such that σ_{1} = ad_{q_{1}} and σ_{2} = ad_{q_{2}}. By Proposition 3.1, it follows from (3) that q = ⟨λ⟩q_{1}q_{2} for some λ ∈ F^{×}, hence q is decomposable if dim V_{1}, dim V_{2} > 1, i.e. if deg A_{1}, deg A_{2} > 1.

If σ_{1} and σ_{2} are symplectic, then they are adjoint to some skew-symmetric bilinear forms b_{1} on V_{1} and b_{2} on V_{2}. By [12, Corollary 6.10], the polar form b_{q} of q satisfies b_{q} ≃ λb_{1} ⊗ b_{2} for some λ ∈ F^{×}, hence q is hyperbolic. Since hyperbolic forms of dimension at least 4 are decomposable, the proof is complete. □

We now use results from Section 2 to derive from (3) the decomposability of q over special fields.

Proposition 3.3. Suppose the adjoint involution ad_{q} decomposes as in (3) with deg A_{1}, deg A_{2} > 1. Then the quadratic form q is decomposable if the field F satisfies any of the following conditions:

• the Brauer group of F is trivial;

• u(F) ≤ 4;

• F is real-closed;

• F is a global field.

Proof. Whenever A_{1} and A_{2} are split—which is certainly the case if the Brauer group of F is trivial—the decomposability of q follows from Corollary 3.2. We may therefore restrict our discussion to the case where A_{1} and A_{2} are not split. The Brauer class of A_{1} and A_{2} then has order 2, hence deg A_{1} and deg A_{2} are even, and therefore dim q ≡ 0 mod 4. Moreover, by [12, Lemma 10.25] or [5, (7.3)], disc q = 1. The decomposability of q follows from Corollary 2.11 if u(F) ≤ 4, from Corollary 2.13 if F is real-closed, and from Corollary 2.16 if F is global. □

The decomposability of q can also be derived from certain types of decompositions of ad_{q}:

Proposition 3.4. Suppose the adjoint involution ad_{q} decomposes as in (3) with deg A_{1}, deg A_{2} > 1. Then the quadratic form q is decomposable in each of the following cases:

(a) A_{1} and A_{2} are Brauer-equivalent to a quaternion F -algebra Q and σ_{1}, σ_{2} are symplectic;

(b) deg A_{1} = 2 and σ_{1}, σ_{2} are orthogonal;

(c) deg A_{1} = 4 and σ_{1}, σ_{2} are symplectic;

(d) deg A_{1} = 4, σ_{1}, σ_{2} are orthogonal and disc σ_{1} = 1;

(e) deg A_{1} = 8, σ_{1}, σ_{2} are symplectic and A_{1} is not a division algebra;

(f) deg A_{1} = 8, σ_{1}, σ_{2} are orthogonal, disc σ_{1} = 1 and one of the factors of the

Clifford algebra C(A_{1}, σ_{1}) is split. Moreover, in case (a) the form q is divisible by the norm form n_{Q} of Q, and in case (b) it is divisible by the form ⟨1, − disc σ_{1}⟩.

Proof. In case (a), we may assume Q is a division algebra, since the proposition follows from Corollary 3.2 if A_{1} and A_{2} are split. Let γ be the conjugation involution on Q. Since σ_{1} and σ_{2} are symplectic, it follows from [12, Proposition, p. 202] or [1, Proposition 3.4] that there exist quadratic forms q′_{1}, q′_{2} on F -vector spaces V′_{1}, V′_{2} such that

(A_{1}, σ_{1}) ≃ (Q, γ) ⊗_{F} (End_{F} V′_{1}, ad_{q′_{1}}), (A_{2}, σ_{2}) ≃ (Q, γ) ⊗_{F} (End_{F} V′_{2}, ad_{q′_{2}}).

On the other hand, by [5, (11.1)], we have

(Q, γ) ⊗_{F} (Q, γ) ≃ (End_{F} Q, ad_{n_{Q}}).

Therefore, (3) implies

(End_{F} V, ad_{q}) ≃ (End_{F} Q, ad_{n_{Q}}) ⊗_{F} (End_{F} V′_{1}, ad_{q′_{1}}) ⊗_{F} (End_{F} V′_{2}, ad_{q′_{2}}).

It follows by [12, Corollary 6.10] that q ≃ ⟨λ⟩n_{Q}q′_{1}q′_{2} for some λ ∈ F^{×}, hence q is decomposable if dim V′_{1} or dim V′_{2} is at least 2. If dim V′_{1} = dim V′_{2} = 1, then q is a multiple of n_{Q} hence it is decomposable by Proposition 2.8.

In case (b), the quaternion algebra with involution (A_{1}, σ_{1}) becomes hyperbolic over F (\sqrt{disc σ_{1}}), hence q also becomes hyperbolic over this extension. By Lemma 2.3, there exists a quadratic form q′ such that

q_{a} ≃ ⟨1, − disc σ_{1}⟩ ⊗ q′.

Since disc q = 1 by [12, Lemma 10.25] or [5, (7.3)], the dimension of q′ is even. Therefore, the Witt index i_{W} (q) is even. Letting i_{W} (q) = 2m, we have

q = q_{a} ⊥ 2m⟨1, −1⟩ ≃ ⟨1, − disc σ_{1}⟩ ⊗ (q′ ⊥ m⟨1, −1⟩).

Cases (c)—(f) reduce to (a) or (b) since in each case there is a decomposition of the form

(A_{1}, σ_{1}) ≃ (Q, σ) ⊗_{F} (A′, σ′)

for some quaternion F -algebra Q: see [12, Proposition 10.21] for case (c), [12, Proposition 10.26] for case (d), [2, Theorem 7] for case (e) and [5, (42.11)] for case (f). □

3.2. Negative results on Shapiro’s question. For any central simple algebra with orthogonal or symplectic involution (A, σ) over a field F, the involution trace form T_{σ}: A → F is defined by

T_{σ} = Trd_{A}(σ(x)x),

where Trd_{A} is the reduced trace. By [5, (11.1)], the map σ_{∗}: A ⊗_{F} A → End_{F} A defined by σ_{∗}(a ⊗ b)(x) = axσ(b) for a, b, x ∈ A carries the involution σ ⊗ σ on A ⊗_{F} A to the adjoint involution of T_{σ}, so

(End_{F} A, ad_{T_{σ}}) ≃ (A, σ) ⊗ (A, σ).

Therefore, ad_{T_{σ}} is decomposable if deg A > 1. However, we show in this subsection that the quadratic form T_{σ} is indecomposable for certain orthogonal involutions σ on central simple algebras A of degree deg A ≡ 2 mod 4, providing a negative solution to Shapiro’s question.

Remark. If the base field F is a global field (or a real-closed field, or satisfies u(F) ≤ 4), Proposition 3.3 shows that the form T_{σ} is decomposable when deg A > 1, since its adjoint involution is decomposable.

We start our construction of examples where T_{σ} is indecomposable with a lemma from elementary number theory.

Lemma 3.5. Suppose m, k, l, r, s ≥ 1 are integers such that

m^{2} = kl, (m − 1)^{2} = rs, (4)

k ≥ r ≥ 1, l ≥ s ≥ 1. (5)

If m ≡ s ≡ 1 mod 2, then l = 1.

Proof. The hypotheses show that l is odd. Let k − r = a and l − s = 2b. By (5), we have k > a ≥ 0 and l > 2b ≥ 0, hence, by the first equation in (4),

(6)  m^{2} > 2ab ≥ 0.

By definition of a and b,

2ab = (k − r)(l − s) = (k − r)l + k(l − s) + rs − kl,

hence, by (4),

2ab = al + 2bk + (m − 1)^{2} − m^{2}.

It follows that

al + 2bk = 2m − 1 + 2ab,

hence, using the first equation in (4),

(al − 2bk)^{2} = (al + 2bk)^{2} − 8abkl = (2m − 1 + 2ab)^{2} − 8abm^{2}.

This equation shows that

(2m − 1 + 2ab)^{2} ≥ 8abm^{2},

hence

(7)  (2ab − 1)(2ab − 1 + 4m − 4m^{2}) ≥ 0.

By (6) we have

2ab − 1 + 4m − 4m^{2} < 4m − 1 − 3m^{2},

and the right side is easily seen to be nonpositive for all m ≥ 1. Therefore, (7) yields 2ab ≤ 1, hence a = 0 or b = 0 since a and b are nonnegative integers. If a = 0, then (4) yields

m^{2} = kl  and  (m − 1)^{2} = ks.

Since m^{2} and (m − 1)^{2} are relatively prime, it follows that k = 1, hence s = (m − 1)^{2} ≡ 0 mod 2, a contradiction. Therefore, b = 0 and (4) implies l = 1 since m^{2} and (m − 1)^{2} are relatively prime. □

Theorem 3.6. Let Q be a quaternion algebra over a field F and let m be an odd integer. If an orthogonal involution σ on the matrix algebra M_{m}(Q) satisfies the following conditions:

(a) the biquaternion algebra Q ⊗_{F} (disc σ, −1)_{F} is division,

(b) there is an ordering P on F such that sgn_{P} T_{σ} = 4(m − 1)^{2}, then T_{σ} is an indecomposable form.

Proof. Suppose T_{σ} = q_{1}q_{2} for some quadratic forms q_{1}, q_{2}. We use condition (a) to show that dim q_{1} and dim q_{2} cannot be both even, and condition (b) to show that the decomposition is trivial if one of dim q_{1}, dim q_{2} is odd.

Suppose first dim q_{1} ≡ dim q_{2} ≡ 0 mod 2. By Lemma 2.1, the Clifford invariant c(q_{1}q_{2}) has index at most 2. On the other hand, the Hasse invariant of T_{σ} was computed by Lewis in [6] and Quéguiner in [10]. From their computation, and from the relations between the Hasse and the Clifford invariant (see [11, p. 81]), it follows that c(T_{σ}) is represented by the biquaternion algebra Q⊗_{F} (disc σ, −1)_{F}, which has index 4 if condition (a) holds. Therefore, this case leads to a contradiction.

Suppose next that dim q_{2} is odd. Since dim T_{σ} = 4m^{2}, the dimension of q_{1} is a multiple of 4. Let dim q_{1} = 4k and dim q_{2} = l, so

m^{2} = kl, k, l, m ≥ 1 and m ≡ 1 mod 2.

Comparing signatures at P, we also have

sgn_{P} T_{σ} = 4(m − 1)^{2} = sgn_{P} q_{1} sgn_{P} q_{2},

with sgn_{P} q_{1} ≡ dim q_{1} ≡ 0 mod 2 and sgn_{P} q_{2} ≡ dim q_{2} ≡ 1 mod 2. Therefore, sgn_{P} q_{1} is a multiple of 4. Letting | sgn_{P} q_{1}| = 4r and | sgn_{P} q_{2}| = s, we have

(m − 1)^{2} = rs, r, s ≥ 1 and s ≡ 1 mod 2.

Lemma 3.5 shows that l = 1, so the decomposition T_{σ} = q_{1}q_{2} is trivial. □

Let be the conjugation involution on a quaternion division F -algebra Q. For x ∈ M_{m}(Q), we define \overline{x} by letting act on each entry of x, so the map x ↦ \overline{x}^{t} is a symplectic involution on M_{m}(Q). To obtain orthogonal involutions, consider diagonal matrices of pure quaternions

Δ = diag(α_{1}, …, α_{m})

with \overline{α_{i}} = −α_{i} ≠ 0 for i = 1, …, m, and define σ_{Δ} : M_{m}(Q) → M_{m}(Q) by

(8)  σ_{Δ}(x) = Δ^{−1} · \overline{x}^{t} · Δ  for x ∈ M_{m}(Q).

Since \overline{Δ}^{t} = −Δ, the map σ_{Δ} is an orthogonal involution, see [5, (2.7)]. Moreover, since every orthogonal involution on the endomorphism algebra of a finitedimensional Q-vector space is adjoint to some skew-hermitian form with respect to (see [5, (4.2)]), every orthogonal involution on M_{m}(Q) is conjugate to an involution σ_{Δ} for some Δ as above.

Proposition 3.7. The discriminant of σ_{Δ} and the signature of T_{σ_{Δ}} at an ordering P of F are determined as follows:

(a) disc σ_{Δ} = α_{1}^{2} … α_{m}^{2}F^{×2} ∈ F^{×}/F^{×2}.

(b) Let F_{P} be a real closure of F at P. If Q is not split by F_{P}, then

sgn_{P} T_{σ_{Δ}} = 0.

If Q is split by F_{P}, choose a pure quaternion β ∈ Q^{×} such that β^{2} <_{P} 0, let m_{1} be the number of indices i such that α_{i}^{2} <_{P} 0 and Trd_{Q}(α_{i}β) >_{P} 0, and let m_{2} be the number of indices i such that α_{i}^{2} <_{P} 0 and Trd_{Q}(α_{i}β) <_{P} 0; then

sgn_{P} T_{σ_{Δ}} = 4(m_{1} − m_{2})^{2}.

Proof. The arguments^{1} in the proof of [5, (7.3)(2)] show that

disc σ_{Δ} = (−1)^{m} Nrd_{M_{m}(Q)}(Δ)F^{×2} = (−1)^{m} Nrd_{Q}(α_{1}) … Nrd_{Q}(α_{m})F^{×2}.

Part (a) follows, since α_{i}^{2} = − Nrd_{Q}(α_{i}).

If Q is not split by F_{P}, then sgn_{P} T_{σ_{Δ}} = 0 by [7, Theorem 1] (or [5, (11.11)]). For the rest of the proof, suppose Q is split by F_{P}. For x = (x_{ij})_{1≤i,j≤m} ∈ M_{m}(Q) we have σ_{Δ}(x) = (x′_{ij})_{1≤i,j≤m} where

x′_{ij} = α_{i}^{−1}\overline{x_{ji}}α_{j}.

Therefore,

T_{σ_{Δ}}(x) = ∑_{i,j=1}^{m} Trd_{Q}(α_{i}^{−1}\overline{x_{ji}}α_{j}x_{ji}).

Letting T_{ij}: Q → F denote the quadratic form

T_{ij}(x) = Trd_{Q}(α_{i}^{−1}\overline{x}α_{j}x),

we thus have

T_{σ_{Δ}} = ⨁_{1≤i,j≤m}^{⊥} T_{ij},

1There is a misprint in the statement of [5, Proposition (7.3)(2)]: the formula given there yields det(Int(u) ∘ σ) instead of disc(Int(u) ∘ σ).

hence

sgn_{P} T_{σ_{Δ}} = ∑_{i,j=1}^{m} sgn_{P} T_{ij}.

If α_{i} and α_{j} anticommute, then 1 and α_{i} span a totally isotropic subspace of Q for T_{ij}, so T_{ij} is hyperbolic and sgn_{P} T_{ij} = 0. Note that in this case we have either α_{i}^{2} >_{P} 0 or α_{j}^{2} >_{P} 0 since Q splits over F_{P}. If α_{i} and α_{j} do not anticommute, pick a nonzero quaternion α′ which anticommutes with α_{i}. Computation shows that (1, α_{i}, α_{j}α′, α_{j}α_{i}α′) is an orthogonal basis of Q for T_{ij}, and that T_{ij} has the following diagonalization in this basis:

T_{ij} = ⟨T (α_{i}^{−1}α_{j})⟩⟨1, −α_{i}^{2}, −α_{j}^{2}α′^{2}, α_{i}^{2}α_{j}^{2}α′^{2}⟩.

Note that α′^{2} >_{P} 0 if α_{i}^{2} <_{P} 0 since Q splits over F_{P}. It follows that sgn_{P} T_{ij} = 0 unless α_{i}^{2} <_{P} 0 and α_{j}^{2} <_{P} 0; in this case, sgn_{P} T_{ij} = ±4. Therefore, letting

t_{ij} = \begin{cases}0 & if α_{i}^{2} >_{P} 0 or α_{j}^{2} >_{P} 0, \\ 1 & if α_{i}^{2} <_{P} 0, α_{j}^{2} <_{P} 0 and Trd_{Q}(α_{i}^{−1}α_{j}) >_{P} 0, \\ −1 & if α_{i}^{2} <_{P} 0, α_{j}^{2} <_{P} 0 and Trd_{Q}(α_{i}^{−1}α_{j}) <_{P} 0, \end{cases}

we have

sgn_{P} T_{σ_{Δ}} = 4 ∑_{i,j=1}^{m} t_{ij}.

Now, let β ∈ Q^{×} be a pure quaternion such that β^{2} <_{P} 0. If a pure quaternion α ∈ Q satisfies Trd_{Q}(αβ) = 0, then β anticommutes with α, hence α^{2} >_{P} 0 since Q splits over F_{P}. For i = 1, …, m, let

s_{i} = \begin{cases}0 & if α_{i}^{2} >_{P} 0, \\ 1 & if α_{i}^{2} <_{P} 0 and Trd_{Q}(α_{i}β) >_{P} 0, \\ −1 & if α_{i}^{2} <_{P} 0 and Trd_{Q}(α_{i}β) <_{P} 0, \end{cases}

so that

∑_{i=1}^{m} s_{i} = m_{1} − m_{2}.

To complete the proof, we have to show

∑_{i,j=1}^{m} t_{ij} = (∑_{i=1}^{m} s_{i})^{2}.

It clearly suffices to prove t_{ij} = s_{i}s_{j} for i, j = 1, …, m, which amounts to showing that if α_{i}^{2} <_{P} 0 and α_{j}^{2} <_{P} 0, then Trd_{Q}(α_{i}^{−1}α_{j}) >_{P} 0 if and only if Trd_{Q}(α_{i}β) and Trd_{Q}(α_{j}β) have the same sign. Since α_{i}^{2} = − Nrd_{Q}(α_{i}) and Trd_{Q}(α_{i}α_{j}) = α_{i}^{2} Trd_{Q}(α_{i}^{−1}α_{j}), it is equivalent to show that if Nrd_{Q}(α_{i}) >_{P} 0 and Nrd_{Q}(α_{j}) >_{P} 0, then

Trd_{Q}(α_{i}β) Trd_{Q}(α_{j}β) Trd_{Q}(α_{i}α_{j}) <_{P} 0.

To check this statement, we may extend scalars to a real closure of F at P. We may therefore substitute for Q a matrix algebra over a real-closed field. The following lemma completes the proof:

Lemma 3.8. Let K be an ordered field and let α, α′, β ∈ M_{2}(K) be matrices with trace zero. If det α, det α′, det β > 0, then tr(αα′) tr(αβ) tr(α′β) < 0.

Proof. Since tr(α) = tr(β) = 0,

tr(αβ) = αβ + βα.

Therefore, letting α_{0} = α − βαβ^{−1}, we have

(9)  2α = tr(αβ)β^{−1} + α_{0}  and  tr(α_{0}β^{−1}) = 0.

The condition det α > 0 then yields

(10)  tr(αβ)^{2} det β^{−1} + det α_{0} > 0.

Similarly, letting α′_{0} = α′ − βα′β^{−1},

(11)  2α′ = tr(α′β)β^{−1} + α′_{0},  tr(α′_{0}β^{−1}) = 0,

and

(12)  tr(α′β)^{2} det β^{−1} + det α′_{0} > 0.

By (9) and (11),

4 tr(αα′) = tr(α_{0}α′_{0}) + tr(αβ) tr(α′β) tr(β^{−2}).

Since tr(β) = 0, we have β^{2} = − det β, hence the preceding equation, together with (10) and (12), yields

4 tr(αα′) tr(αβ) tr(α′β) = tr(α_{0}α′_{0}) tr(αβ) tr(α′β) − 2 tr(αβ)^{2} tr(α′β)^{2} det β^{−1}
< tr(α_{0}α′_{0}) tr(αβ) tr(α′β) + tr(αβ)^{2} det α′_{0}
+ tr(α′β)^{2} det α_{0}.

Since tr(α_{0}) = tr(α′_{0}) = 0, the right side of the last inequality is

det(tr(αβ)α′_{0} − tr(α′β)α_{0}).

To conclude the proof, it suffices to observe that the determinant of any matrix which anticommutes with β is negative, otherwise M_{2}(K) would be generated by two anticommuting matrices with negative square, i.e. (−1, −1)_{K} ≃ M_{2}(K). □

Remark. It follows from [7, Theorem 1] that the signature of the involution trace form T_{σ} is a square for any orthogonal or symplectic involution σ on a central simple algebra A. In the terminology of [7] (see also [5, (11.10)]), Proposition 3.7(b) states that if Q is split by F_{P}, then

sgn_{P} σ_{Δ} = |m_{1} − m_{2}|.

We proceed to give examples satisfying the hypotheses of Theorem 3.6, for any odd integer m ≥ 3.

Let F_{0} be an arbitrary ordered field and let F = F_{0}(x, y), the field of rational fractions in two indeterminates over F_{0}. The following classical construction (see [8, p. 75]) extends the ordering on F_{0} to an ordering P on F such that x <_{P} 0 and 0 <_{P} y <_{P} 1: consider the (x, y)-adic valuation on F defined on F_{0}[x, y] \ {0} by

v(∑_{i,j} a_{ij}x^{i}y^{j}) = min{(i, j) | a_{ij} ≠ 0},

where the minimum is for the lexicographic order on ℤ^{2}. If f ∈ F^{×} satisfies v(f) = (m, n), then v((−x)^{−m}y^{−n}f) = 0 and we may consider the residue \overline{(−x)^{−m}y^{−n}f} ∈ F_{0}. Set

(13)  f >_{P} 0  if and only if  \overline{(−x)^{−m}y^{−n}f} > 0 in F_{0}.

Then −x >_{P} 0, y >_{P} 0 and 1 − y >_{P} 0, so x <_{P} 0 and 0 <_{P} y <_{P} 1.

Consider the quaternion F -algebra Q = (x, y)_{F}. For any odd integer m ≥ 3, consider the following pure quaternions in Q:

α_{1} = i + k, α_{2} = j, α_{3} = ⋯ = α_{m} = i,

and let σ_{Δ} be the involution on M_{m}(Q) defined as in (8). Since y >_{P} 0, the algebra Q is split by any real closure of F at P. To compute sgn_{P} T_{σ_{Δ}}, we use Proposition 3.7 with β = i. Since

α_{1}^{2} = x(1 − y) <_{P} 0, α_{2}^{2} = y >_{P} 0 and Trd_{Q}(i(i + k)) = Trd_{Q}(i^{2}) = 2x <_{P} 0,

we have, in the notation of Proposition 3.7, m_{1} = 0 and m_{2} = m − 1, so

sgn_{P} T_{σ_{Δ}} = 4(m − 1)^{2}.

By Proposition 3.7, we also have

disc σ_{Δ} = α_{1}^{2} … α_{m}^{2}F^{×2} = (x − xy)yx^{m−2}F^{×2} = (1 − y)yF^{×2}.

To see that the hypotheses of Theorem 3.6 hold, it remains to prove that the tensor product

(x, y)_{F} ⊗_{F} (y(1 − y), −1)_{F}

is a division algebra, or, equivalently by a theorem of Albert (see [5, (16.5)]), that an associated Albert form such as

q = ⟨x, y, −xy, y(y − 1), y(y − 1), 1⟩

is anisotropic. Since

q = ⟨1, y, y(y − 1), y(y − 1)⟩ ⊥ ⟨x⟩⟨1, −y⟩,

a degree argument (see for instance [4, Lemma 1.4(i)]) shows that q is isotropic over F if and only if one of the forms ⟨1, y, y(y − 1), y(y − 1)⟩, ⟨1, −y⟩ is isotropic over F_{0}(y). The latter is clearly anisotropic since y ∉ F_{0}(y)^{×2}. Using the (y − 1)-adic valuation on F_{0}(y), one may construct as above (see (13)) an ordering P_{0} on F_{0}(y) such that y >_{P_{0}} 1. Then

sgn_{P_{0}}⟨1, y, y(y − 1), y(y − 1)⟩ = 4,

so ⟨1, y, y(y − 1), y(y − 1)⟩ is anisotropic over F_{0}(y).

Thus, Theorem 3.6 shows that the quadratic form T_{σ_{Δ}} is indecomposable, even though its adjoint involution is decomposable, as it is isomorphic to σ_{Δ} ⊗ σ_{Δ}.

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Department of Mathematics, University College Dublin, Belfield, Dublin 4, Ireland

E-mail address: david.lewis@ucd.ie
E-mail address: mohammad.mahmoudi@ucd.ie

Institut de Mathématique Pure et Appliquée, Université catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium

E-mail address: tignol@math.ucl.ac.be
URL: http://www.math.ucl.ac.be/membres/tignol

Department of Mathematics, Eidgenössische Technische Hochschule, CH-8092 Zürich, Switzerland

E-mail address: oliver.villa@math.ethz.ch