When is Galois cohomology free or trivial?

TL;DR

This paper characterizes when Galois cohomology groups are free or trivial over certain field extensions, introduces notions of cohomological freeness and triviality, and provides examples with specified properties.

Contribution

It provides precise conditions for the freeness or triviality of Galois cohomology groups in cyclic extensions and introduces new concepts analogous to cohomological dimension.

Findings

Conditions for H^n(E) to be free or trivial are established.

Inheritance properties of these conditions for higher cohomology groups are analyzed.

Examples with prescribed cohomological dimension are constructed.

Abstract

Let p be a prime and F a field containing a primitive pth root of unity. Let E/F be a cyclic extension of degree p and G_E < G_F the associated absolute Galois groups. We determine precise conditions for the cohomology group H^n(E)=H^n(G_E,Fp) to be free or trivial as an Fp[Gal(E/F)]-module. We examine when these properties for H^n(E) are inherited by H^k(E), k>n, and, by analogy with cohomological dimension, we introduce notions of cohomological freeness and cohomological triviality. We give examples of H^n(E) free or trivial for each n in N with prescribed cohomological dimension.