In his 1976 proof of the converse to Herbrand’s theorem, Ribet used Eisenstein-cuspidal congruences to produce unramified degree-$p$
extensions of the $p$
-th cyclotomic field when $p$
is an odd prime. After reviewing Ribet’s strategy, we will discuss recent work with Preston Wake in which we apply similar techniques to produce unramified degree-$p$
extensions of $\mathbb{Q}(N^{1/p})$
when $N$
is a prime that is congruent to $-1$
mod $p$
. This answers a question posted on Frank Calegari’s blog.