Abstract
A primary problem in the theory of quadratic forms over a field F of characteristic different from two is to prove that the rings Hq*F and GWF are isomorphic. Here HqF = H*(Gal(Fq/F),Z/2Z)), where Fq is the quadratic closure of F, and GWF is the graded Witt nrng associated to the fundamental ideal of even dimensional forms in the Witt ring WF of F. In this paper, we assume we are given a field extension K of F such that WK is 'close' to WF or Hq*K is 'close' to Hq F . A method is developed to obtain information about these graded rings over F and its 2-extensions from information about the corresponding graded ring of K. This relative theory extends and includes the previously developed absolute case where K = Fq. Applications are also given to show that Hq F and GWF are isomorphic for a collection of fields anrsing naturally from the theory of abstract Witt rings.
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