2024-03-182024-03-182022-10-312330-1511http://hdl.handle.net/20.500.12613/9991We show that for primes $N, p \geq 5$ with $N \equiv -1 \bmod p$, the class number of $\mathbb {Q}(N^{1/p})$ is divisible by $p$. Our methods are via congruences between Eisenstein series and cusp forms. In particular, we show that when $N \equiv -1 \bmod p$, there is always a cusp form of weight $2$ and level $\Gamma _0(N^2)$ whose $\ell$th Fourier coefficient is congruent to $\ell + 1$ modulo a prime above $p$, for all primes $\ell$. We use the Galois representation of such a cusp form to explicitly construct an unramified degree-$p$ extension of $\mathbb {Q}(N^{1/p})$.17 pagesengAttribution-NonCommercial-NoDerivs CC BY-NC-NDhttps://creativecommons.org/licenses/by-nc-nd/4.0/Eisenstein idealClass groupGalois representation.Text