The graded Witt ring and Galois cohomology. II

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 $H_q^\ast F$ and $GWF$ are isomorphic. Here $H_q^\ast F = {H^\ast }(\operatorname {Gal}({F_q}/F),{\mathbf {Z}}/2{\mathbf {Z}}))$, where ${F_q}$ is the quadratic closure of $F$, and $GWF$ is the graded Witt ring 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 $H_q^{\ast } K$ is ’close’ to $H_q^\ast F$. 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 $K = {F_q}$. Applications are also given to show that $H_q^\ast F$ and $GWF$ are isomorphic for a collection of fields arising naturally from the theory of abstract Witt rings.