A modular construction of unramified $p$-extensions of $\mathbb{Q}(N^{1/p})$

Jaclyn Lang, Preston Wake

We show that for primes $N,p\ge 5$ with $N\equiv -1\mathrm{mod}p$, the class number of $\mathbb{Q}(N^{1/p})$ is divisible by $p$. Our methods are via congruences between Eisenstein…