Sur le discriminant et la forme trace d'une involution orthogonale en degr� 4

Abstract

let A be a central simple algebra of degree 4 over a fieldF of caracteristic not 2 and \(\sigma\) an orthogonal involution on A. Denotes \(T_\sigma\) the trace form of \((A,\sigma)\). In this note we show that \(T_\sigma\in I^4 F \), ie. \(T_\sigma\) is similar to a 4-Pfister form (over F) if and only if \(disc \sigma \) is a sum of two squares in F. It follows, in the split case , that if \(T_\sigma\in I^4 K\) then, the hermitian level of \((A,\sigma)\) is finite if and only if \(T_\sigma\) is weakly isotropic. A similar result holds with ordinary squares and trace form for any central simple algebra (see [5] and [7]).