Quadratic forms over dyadic valued fields. I. The graded Witt ring

Abstract

This paper gives a detailed account of the arithmetic of quadratic forms over a field F of characteristic 0, carrying a 2-Henselian discrete valuation with residue field of characteristic 2. We give an analogue of Springer's Theorem for the graded Witt ring of such a field, and describe new counterexamples to the amenabilitiy problem for multiquadratic extensions. The sequel to this paper will contain an axiomatic approach to the results contained herein, and will treat the Galois cohomology of such fields.