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
$N$ is a prime that is congruent to
$p$. This answers a question posted on Frank Calegari’s blog.