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We show that for primes $N, p \geq 5$ with $N \equiv -1 \bmod p$, the classnumber of $\mathbb{Q}(N^{1/p})$ is divisible by $p$. Our methods are viacongruences between Eisenstein series and cusp forms. In particular, we showthat when $N \equiv -1 \bmod p$, there is always a cusp form of weight $2$ andlevel $\Gamma_0(N^2)$ whose $\ell$-th Fourier coefficient is congruent to $\ell+ 1$ modulo a prime above $p$, for all primes $\ell$. We use the Galoisrepresentation of such a cusp form to explicitly construct an unramified degree$p$ extension of $\mathbb{Q}(N^{1/p})$.

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