In his 1976 proof of the converse of 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 posed on Frank Calegari’s blog.

https://upenn.zoom.us/j/93547156018?pwd=OEg0enZ2cU8zSkdJVnNVTGJiMXg4QT09

Meeting ID: 935 4715 6018

Passcode: 209