Witt invariants of involutions of low degree in characteristic 2 (Q6633615)

Given a central simple algebra \(A\) of degree \(2m\) over a field \(F\) with symplectic involution \(\sigma\), each element \(a\) in the subspace \(\text{Symd}(\sigma)=\{x+\sigma(x) : x\in A\}\) has characteristic polynomial that has a polynomial square root in \(F[x]\) called the Pfaffian and denoted \(\text{Prp}_{\sigma,a}(X)\) of degree \(m\). The coefficient of \(X^{m-2}\) in the Pfaffian, denoted by \(\text{Srp}_{\sigma}\) is a quadratic form on the space \(\text{Symd}(\sigma)\). The main objective of this paper is to analyse the structure of this quadratic form and relate it to certain properties of \(A\). \N\NTheorem 1 states that if \(\deg A=8\) and \(\text{char}(F)=2\), then there exist a 3-fold Pfister form \(\pi_3\) and a 5-fold Pfister form \(\pi_5\) such that \(\text{Srp}_\sigma=[1,1]\perp \pi_3 \perp \pi_5\). Moreover, \(\pi_5\) is a multiple of \(\pi_3\), \(\pi_3\) is hyperbolic if and only if \(A\) decomposes as a tensor product of three quaternion algebras stable under \(\sigma\), and \(\pi_5\) is hyperbolic if and only if \(\text{Symd}_\sigma\) contains a noncentral element whose square is central. \N\NUsing similar technique, the author proves also Theorem 12 which states that if \(B\) is an Azumaya algebra of degree \(4\) over a quadratic étale extension \(Z\) of a field \(F\) with \(\text{char}(F)=2\) and \(\tau\) is a unitary involution fixing \(F\), \(\text{Sym}_\tau\) is the subspace of elements fixed under \(\tau\) and \(\text{Srd}\) the coefficient of \(X^2\) in the characteristic polynomial, then there exist a 2-fold Pfister form \(\pi_2\) and a 4-fold Pfister form \(\pi_4\) such that \(\text{Srd}_\tau=[1,1]\perp \pi_2 \perp \pi_4\). Moreover, \(\pi_4\) is a multiple of \(\pi_2\), \(\pi_2\) is hyperbolic if and only if \(B\) decomposes as a tensor product of two quaternion algebras stable under \(\tau\), and \(\pi_4\) is hyperbolic if and only if \(\text{Sym}_\tau\) contains a noncentral element whose square is central. The author also provides a statement of a similar flavour about algebras of degree 4 over fields of characteristic two with orthogonal involution.