A Galois-type Springer theorem for hermitian and generalized quadratic forms

Abstract

We prove that anisotropy of nonsingular hermitian and skew-hermitian forms over central simple algebras with involution is preserved under scalar extension to every finite Galois extension of odd degree coprime to the index of the algebra. In characteristic two, the same conclusion holds for generalized quadratic forms over algebras with involution of the first kind, with no nonsingularity assumption on their polar forms. The main tool is an isometry-level descent theorem. For a cyclic extension of odd prime degree, every anisotropic nonsingular form over a division algebra with involution, of rank smaller than the extension degree and with Galois-invariant isometry class, descends to an anisotropic form over the base field. Unlike descent results for Witt classes, this theorem descends the form itself. A bounded-rank reduction, based on injectivity of the Witt-group restriction map, derives the cyclic anisotropy theorem from this descent result; a regularization argument removes the nonsingularity assumption on polar forms in characteristic two, and the Feit-Thompson theorem extends the result to arbitrary finite Galois extensions of odd degree.