Witt’s theorems for Galois Ring valued quadratic forms

Abstract

We prove Witt’s cancelation and extension theorems for Galois Ring valued quadratic forms. The proof is based on the properties of the invariant I , previously defined by the authors, that classifies, together with the type of the corresponding bilinear form (alternating or not), nonsingular Galois Ring valued quadratic forms. Our results extend the Witt’s theorem for mod four valued quadratic forms. On the other hand, the known relation between the invariant I and the Arf invariant of an ordinary quadratic form (if the associated nonsingular bilinear form is alternating) is extended to the nonalternating case by explaining the invariant I in terms of Clifford algebras.