Is there something I am missing about the computation of the $p$-part of the class groups of cyclotomic fields?

Question

Well, the answer of the question in the title in certainly Yes, many things in fact, but let me be more precise.

In 1958, Serre gave a Bourbaki talk on the recent works of Iwasawa on class groups in towers of cyclotomic extensions, in which he presented Iwasawa's theorems on the growth of $\operatorname{Cl}(\mathbb{Z}[\zeta_{p^{n}}])[p^{\infty}]$ in terms of what we nowadays call Iwasawa's $\lambda,\mu$ and $\nu$ invariants. At the end of the introduction, Serre writes that the $\mu$-invariant has been computed (or so it seems, he adds a bit mysteriously) for $p=37,59$ and $67$, the first three irregular primes, which are the first three primes for which these invariants cannot all be trivial.

70 years later and with the help of the Iwasawa Main Conjecture, it is quite easy to carry such a computation, and trivial to do so with electronic aid: one computes a bunch of Bernoulli numbers, deduce from that an approximation of the $p$-adic $L$-function. The Iwasawa Main Conjecture (proved by Mazur-Wiles) then translates this approximate computation to structure properties of $\operatorname{Cl}(\mathbb{Z}[\zeta_{p^{n}}])[p^{\infty}]$. Relying on both, computing $\lambda,\mu$ and $\nu$ for smallish irregular primes is a matter of seconds.

However, even with quite competent electronic aid, trying to compute these invariants directly (that is to say for direct naïve computation of the structure of the class groups in the towers, and by naïve here I mean using the best general algorithms, rather than anything particular to the $p$-part of the class group of the particular cyclotomic field $\mathbb{Q}(\zeta_{p^n})$) strains my computing resources (even checking that the order of the class group of $\mathbb{Q}(\zeta_{67})$ is divisible only once by $67$ is not something my computer seems to find particularly easy).

I suspect that two main differences between what I'm telling my computer to do and what Iwasawa (and/or others) did then is that 1) I am trying to compute the full structure of the class group and not just the $p$-torsion part and 2) I am trying to establish the full structure, and not just proving that $\mu$ vanishes.

Nevertheless, I would be glad to know the answer to the following question

How did Iwasawa himself (or others) compute the structure or at least the structure of the $p$-torsion-part of the class groups of say $\mathbb{Q}(\zeta_{59^n})$ or $\mathbb{Q}(\zeta_{157^n})$ for small $n$ back in the 1950s, since apparently he (or others) knew enough about them to deduce something about $\mu$?

Answer 1

[Rather than leaving the comment "Class number formula" for Olivier as a comment, I expand it for other readers of the question, to a partial answer.]

Kummer knew in 1850 that the class group of $\mathbb{Q}(\zeta_p)$ is divisible by $p$ exactly if $p$ divides a numerator of a Bernoulli number $B_i$ for even $2\leq i\leq p-3$. The reason Iwasawa and others knew a lot about the class numbers of cyclotomic fields (and not just its $p$-part) is that it was studied a lot, not explicitly, but through the class number formula. (I know too little about the history, but Rosen's text in Cornell-Silverman-Stevens is a good source.)

Let $h$ be the class number of $K=\mathbb{Q}(\zeta_m)$ for an integer $m$ (or more generally of an abelian extension of $\mathbb{Q}$) and let $h^+$ be the class number of the maximal real subfield. Then $h$ is the product of $h^+$ and $Q\,w\, \prod_{\chi\text{ odd}} (-\tfrac{1}{2} B_{1,\chi})$ where $w$ is the number of roots of unity in $K$, the integer $Q$ is $1$ if $m$ is a prime power, and $B_{1,\chi}$ are Bernoulli numbers. Then $h^+$ can be expressed as the index of the cyclotomic units inside the full group of global units. Also $h/h^+=h^-$ is the index of the Sitckelberger elements in the group algebra. These were the methods to calculate $h$. See Chapter 3 in Lang's Cyclotomic Fields. This is all before Iwasawa's theorem.

If you are trying to calculate the actual class group of a field of moderately large degree, like 66, the Minkowski bound (or any improvement on it) is huge and it is practically impossible to do this. The difference is that you are trying to exhibit actual elements in the class group rather than its order. I believe you can also find more information about the structure of the $p$-primary class group from Bernoulli numbers, like Herbrand's theorem does; but that is certainly better explained now through the main conjecture. But, I believe, you won't get the elements in the class group from them.

Answer 2

Recently, I stumbled coincidentally on the paper

Computation of invariants in the theory of cyclotomic fields K. Iwasawa and C. Sims J. Math. Soc. Japan Vol.18 (1966)

This explains in full details how Iwasawa and Sims carried over the computation of my question for primes up to $4001$ (so way beyond what I had envisioned). It is clear from the theoretical method they describe there that even without electronic aid, smallish regular primes are accessible, especially since D.H. Lehmer, Emma Lehmer and Henry Vandiver had already used the SWAC to compute huge table of Bernoulli numbers in 1954.

In addition, Iwasawa proved a criterion in terms of congruences with Bernoulli numbers in

On Some Invariants of Cyclotomic Fields K. Iwasawa American Journal of Mathematics, Vol. 80, No. 3 (1958)

for the vanishing of the $\mu$-invariant. Using this criterion, it is extremely easy to obtain just what Serre says was apparently computed, that is to say the vanishing of the $\mu$-invariant for the first three irregular primes. Since this result is almost contemporary with Serre's talk, I think that the answer to my question is: Serre had heard of this paper of Iwasawa but possibly did not know its precise content.