Nicole Lemire, Jan Minac, John Swallow: Galois module structure of Galois cohomology,,

Submission: 2004, Oct 30

Let $F$ be a field containing a primitive $p$th root of unity, and let $U$ be an open normal subgroup of index $p$ of the absolute Galois group $G_F$ of $F$. We determine the structure of the cohomology group $H^n(U,\Fp)$ as an $\Fp[G_F/U]$-module for all $n\in\mathbb{N}$. Previously this structure was known only for $n=1$, and until recently the structure even of $H^1(U,\Fp)$ was determined only for $F$ a local field, a case settled by Borevi\v{c} and Faddeev in the 1960s.

2000 Mathematics Subject Classification: 12G05; 19D45

Keywords and Phrases: Galois cohomology, Milnor K-theory, norm map, Hilbert 90, cyclic extension, Galois module

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