On the computation of trace forms of algebras with involution

Definitions and notation: Let A be a central simple algebra of degree n over a field k of characteristic different from 2. An involution on A is a ring antiautomorphism of order at most 2. An involution σ is of the first kind if σ|k = Idk, and of the second kind if σ|k is a non trivial involution on k, denoted by .̄ In the last case, k is a quadratic extension of the subfield k0 fixed by .̄ So we have k = k0( √ α), α ∈ k∗ 0/k∗2 0 . The involution ̄ is then defined by u + v √ α = u− v√α, with u, v ∈ k0. If A is split, the involutions of the first kind are exactly the involutions adjoint to symmetric or skew symmetric bilinear forms. In the case where A is a central simple algebra with an involution of the first kind, and if L is a splitting field of A, A ⊗ L is a split algebra and σ ⊗ IdL is an involution on A ⊗ L, then adjoint to a bilinear form b. We say that σ is of orthogonal type if b is symmetric , and of symplectic type if b is skew symmetric. This definition does not depend on the splitting field. Now consider the function (x, y) ∈ A×A 7→ TrdA(σ(x)y). If the involution σ is of the first kind, this is a nondegenerate symmetric bilinear form of dimension n over k, and we denote by Tσ the corresponding quadratic form. If σ is of the second kind, this is a hermitian form over k, denoted by Hσ. Then we define Tσ by Tσ (x) = Hσ (x, x). This is a nondegenerate quadratic form over k0 of dimension 2n . If σ is of the first kind, we set Alt(σ)={ x− εσ(x), x ∈ A }, with ε = 1 if σ is orthogonal, and ε = −1 if σ is symplectic. Recall now the definition of the determinant of an involution of the first kind, given by Knus, Parimala and Sridharan: