  We 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 $\mathbb{Q}(N^{1/p})$.
