Class groups of Kummer extensions via cup products in Galois cohomology

Abstract

We use Galois cohomology to study the $p$-rank of the class group of $\mathbf{Q}(N^{1/p})$, where $N\equiv 1\mathrm{mod}p$ 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 $N$.

54 pages. This version has been accepted for publication in Transaction of the AMS