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

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