The Bloch--Kato conjecture, decomposing fields, and generating cohomology in degree one

TL;DR

This paper explores the structure of Galois cohomology rings, characterizes finite groups with degree one generation property, and introduces the concept of decomposing fields to refine the Bloch--Kato conjecture, supported by explicit computations.

Contribution

It characterizes finite groups with cohomology generated in degree one, introduces decomposing fields for cohomology classes, and refines the Bloch--Kato conjecture through explicit examples.

Findings

Finite groups with cohomology generated in degree one are characterized.

Decomposing fields are defined and studied for cohomology classes.

Explicit cohomology rings are computed for specific fields.

Abstract

The famous Bloch--Kato conjecture implies that for a field $F$ containing a primitive $p$th root of unity, the cohomology ring of the absolute Galois group $G_F$ of $F$ with $\mathbb{F}_p$ coefficients is generated by degree one elements. We investigate other groups with this property and characterize all such groups that are finite. Restricting to the case of $p$-groups, our work answers a question of Quadrelli, Snopce and Vanacci posed in 2022. As a further step in this program, we study implications of the Bloch--Kato conjecture to cohomological invariants of finite field extensions. Conversely, these cohomological invariants have implications for refining the Bloch--Kato conjecture. In service of such a refinement, we define the notion of a decomposing field for a cohomology class of a finite field extension and study minimal decomposing fields of degree two cohomology classes arising from degree $p$ extensions. We illustrate this refinement by explicitly computing the cohomology rings of superpythagorean fields and $p$-rigid fields. Finally, we construct nontrivial examples of cohomology classes and their decomposing fields, which rely on computations by David Benson in the appendix.