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.

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Last updated: 30 May 2023