The structure of class groups of number fields can be computed in some cases with explicit congruence conditions, for example as in Kummer's criterion which relates the p-part of the class group of the $p$-th cyclotomic field to congruences of Bernoulli numbers mod $p$. For $p$ and $N$ prime with $N=1$ mod $p$, a similar result of Calegari and Emerton relates the rank of the $p$-part of the class group of $\mathbb{Q}(N^{1/p})$ to whether or not a certain quantity (Merel's number) is a $p$-th power mod $N$.

I'll speak about joint work with Karl Schaefer in which we study this rank, refining the result of Calegari and Emerton and proving exact characterizations of the rank for small $p$ in terms of similar $p$-th power congruence conditions. Our main tactic is to relate elements of the class group to the vanishing of cup products in Galois cohomology. I'll aim to give some gentle exposition of how to think about these cup products, and I'll highlight the ways in which our work was informed by computation.