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

Sunil K. Chebolu, Ján Mináč, Cihan Okay, Andrew Schultz, Charlotte Ure

TL;DR

The paper develops a refinement program for the Bloch--Kato conjecture by introducing decomposing fields and a filtration tower $F^{(n)}$, linking degree-one data to higher cohomology via inflation and decomposability. It proves finite-quotient refinements, establishes minimal decomposing fields for degree-two cohomology, and provides explicit cohomology calculations for superpythagorean and $p$-rigid fields, including metacyclic-group models and a topological counterpart. It also characterizes finite groups with cohomology generated in degree one, and extends the framework to Bloch--Kato pro-$p$-groups with strong variants, suggesting a broad program to understand indecomposable-to-decomposable behavior under inflation across finite quotients. Collectively, these results deepen the connection between Galois cohomology, Milnor $K$-theory, and group-theoretic structure, offering concrete refinement tools for the Bloch--Kato landscape and new avenues for systematic cohomological analysis.

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. 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.

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

TL;DR

The paper develops a refinement program for the Bloch--Kato conjecture by introducing decomposing fields and a filtration tower , linking degree-one data to higher cohomology via inflation and decomposability. It proves finite-quotient refinements, establishes minimal decomposing fields for degree-two cohomology, and provides explicit cohomology calculations for superpythagorean and -rigid fields, including metacyclic-group models and a topological counterpart. It also characterizes finite groups with cohomology generated in degree one, and extends the framework to Bloch--Kato pro--groups with strong variants, suggesting a broad program to understand indecomposable-to-decomposable behavior under inflation across finite quotients. Collectively, these results deepen the connection between Galois cohomology, Milnor -theory, and group-theoretic structure, offering concrete refinement tools for the Bloch--Kato landscape and new avenues for systematic cohomological analysis.

Abstract

The famous Bloch--Kato conjecture implies that for a field containing a primitive th root of unity, the cohomology ring of the absolute Galois group of with coefficients is generated by degree one elements. We investigate other groups with this property and characterize all such groups that are finite. 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 extensions. We illustrate this refinement by explicitly computing the cohomology rings of superpythagorean fields and -rigid fields. Finally, we construct nontrivial examples of cohomology classes and their decomposing fields, which rely on computations by David Benson in the appendix.
Paper Structure (16 sections, 20 theorems, 105 equations)

This paper contains 16 sections, 20 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. History and basic facts on the Bloch--Kato conjecture
  4. Kummer Theory
  5. Characterizing classes in $H^2(G, \mathbb{F}_p)$
  6. The Galois embedding problem
  7. Quotients of absolute Galois groups
  8. The second cohomology refinement of the Bloch--Kato conjecture
  9. An example of splitting without vanishing
  10. Refining Bloch--Kato: Superpythagorean and $p$-rigid fields
  11. Cohomology of metacyclic groups
  12. Proof of \ref{['thm:example']}
  13. Topological model
  14. Groups whose cohomology ring is generated in degree one
  15. Bloch--Kato pro-p-groups
  16. ...and 1 more sections

Key Result

Theorem 1.1

Suppose that $G$ is a finite group, $p$ is a prime number that divides $|G|$, and $G$ acts trivially on $\mathbb{F}_p$. Then $H^*(G,\mathbb{F}_p)$ is generated in degree one if and only if $p=2$ and the Sylow $2$-subgroup of $G$ is a nontrivial elementary $2$-abelian group that admits a normal compl

Theorems & Definitions (45)

  • Theorem 1.1
  • Definition 1.2
  • Definition 1.3: Decomposing field
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • ...and 35 more