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

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.