Class field towers

Question

It is known (Golod and Shafarevich) that the class field tower of a finite extension $K$ of $\mathbb{Q}$ may be infinite. But is it always finite for $K=\mathbb{Q}[\zeta]$ where $\zeta$ is a root of unity (in particular, when $\zeta^p=1$ where $p$ is a prime)?

Answer 1

Let $\ell$ be an odd prime and $m$ an integer such that $$|\{p|m,\ p\equiv1\operatorname{mod} \ell\}|\geq8.$$ Then Y.Furuta proved that $\mathbb Q(\zeta_m)$ admits an infinite unramified $\ell$-class field tower (Nagoya Math. Journal,1972). In fact, I.Shparlinski proved using this result that $\mathbb Q(\zeta_m)$ admits an infinite class field tower for almost all $m$ so the answer to your first question is maximally negative.

Even when $\zeta_p$ is a primitive $p$-root of unity, $\mathbb Q(\zeta_p)$ can admit infinite $p$-class field tower, as was first shown (I believe) by R.Schoof (Crelle,1986). I don't have access to this article at present but $p=877$ is I believe an example. In fact, it seems likely that infinitely many prime numbers (perhaps even density one) are such that $\mathbb Q(\zeta_p)$ admits an infinite class field tower (see below for the reason).

As for your implicit question about FLT, not only doesn't it work for the reason explain by David Loeffler, it also cannot possibly work philosophically (at present). Indeed, the study of maximal unramified extension of cyclotomic fields is closely linked to Iwasawa theory and the best results known about it (Iwasawa Main Conjecture, McCallum-Sharifi's conjectures...) and for instance the statement above about many $\mathbb Q(\zeta_p)$ having infinite class field towers are obtained by linking this problem with Galois representations of unramified modular forms with residually reducible representation, so that an alternate proof of FLT along these lines would (under current knowledge) not rely on arguments significantly simpler than the arguments of the current proof.