## The graded Witt ring and Galois cohomology. II

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- by Jón Kr. Arason, Richard Elman and Bill Jacob PDF
- Trans. Amer. Math. Soc.
**314**(1989), 745-780 Request permission

## 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.## References

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*Hilbert*$90$

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## Additional Information

- © Copyright 1989 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**314**(1989), 745-780 - MSC: Primary 11E04; Secondary 11E81, 12G05, 18F25, 19G12
- DOI: https://doi.org/10.1090/S0002-9947-1989-0964897-9
- MathSciNet review: 964897