What is the importance of Herbrand-Ribet theorem?

Question

Does Herbrand-Ribet theorem say anything about the Sylow subgroups of the ideal class group of $\mathbb Q(\zeta_p)$ for primes not equal to $p$? Further, it seems only to determine when the Sylow-p-subgroup of the ideal class group is non-zero. Does it give anything specific regarding its size?

Answer 1

Herbrand-Ribet only tells you about the $p$ part of the class group of $\mathbb{Q}(\zeta_p)$. Ribet's proof says a bit more than just when the $p$-part of the classgroup is non-trivial, it establishes an equivalence between $p$ dividing the numerator of the Bernoulli number $B_{p-n}$ and the $\chi^{n}$-part of the class group being nonzero, for odd $n$. (By $\chi^n$-part I mean the part on which $\text{Gal}(\mathbb{Q}(\zeta_p)/\mathbb{Q}$) acts by the $n$-th power of the cyclotomic character.) Note that when $n$ is even, Vandiver's conjecture predicts that the corresponding part of the class group is trivial.

This is all that Ribet-Herbrand says, but in Mazur-Wiles's proof of the Main Conjecture of Iwasawa Theory (in "Class Fields of Abelian Extension of $\mathbb {Q}$"), they establish (as a corollary to their theorem) that in fact the exact power of $p$ dividing the numerator of $B_{p-n}$ is the exact power of $p$ dividing the $\chi^n$ part of the class group, which is an obvious strengthening of Ribet-Herbrand.