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 $N \equiv 1 \bmod {p}$ 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 $N$.