Class groups of Kummer extensions via cup products in Galois cohomology

TL;DR

This paper employs Galois cohomology to analyze the p-rank of class groups in Kummer extensions, providing new insights and characterizations, especially for the case p=5, linking algebraic properties to explicit number-theoretic conditions.

Contribution

It offers a partial converse to Calegari--Emerton's theorem, explains known counterexamples, and fully characterizes the 5-rank of class groups in specific Kummer extensions.

Findings

Proves a partial converse to Calegari--Emerton's theorem.

Provides a new explanation for counterexamples to the full converse.

Characterizes the 5-rank of class groups in terms of explicit power residue conditions.

Abstract

We use Galois cohomology to study the $p$-rank of the class group of $\mathbf{Q}(N^{1/p})$, where $N \equiv 1 \bmod{p}$ is prime. We prove a partial converse to a theorem of Calegari--Emerton, and provide a new explanation of the known counterexamples to the full converse of their result. In the case $p = 5$, we prove a complete characterization of the $5$-rank of the class group of $\mathbf{Q}(N^{1/5})$ in terms of whether or not $\prod_{k=1}^{(N-1)/2} k^{k}$ and $\frac{\sqrt{5} - 1}{2}$ are $5$th powers mod $N$.