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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)?

  • The Golod--Shafarevich examples include cases where $K$ is imaginary quadratic, so $K$ is contained in $\mathbf{Q}(\zeta_n)$ for some suitable $n$; it follows that $\mathbf{Q}(\zeta_n)$ also has infinite class field tower. This doesn't deal with your more specific question about prime-order cyclotomic fields, though. – David Loeffler Apr 23 '15 at 07:09
  • @DavidLoeffler: Would the case when $p$ is prime imply FLT? –  Apr 23 '15 at 07:15
  • Yes, because anything implies a true statement :-). Seriously, why should this question have any particular relation to FLT? – David Loeffler Apr 23 '15 at 07:18
  • @DavidLoeffler: If the class number of $\mathbb{Q}[\zeta]$ is 1, $\zeta^n=1$, then FLT for that $n$ follows, right? Isn't it enough to assume that $\zeta$ is inside a number field with class number 1? –  Apr 23 '15 at 07:21
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    If the class number of $\mathbf{Q}[\zeta]$ is coprime to $p$, where $\zeta$ is a primitive $p$-th root of unity and $p \ge 3$ is prime, then FLT for exponent $p$ follows. But this really needs control of the class group of the cyclotomic field itself; I can't see how knowing that $\mathbf{Q}[\zeta]$ embeds in some other larger field of class number 1 helps in any way. – David Loeffler Apr 23 '15 at 11:26
  • I very much doubt that FLT would come into reach even if the class number of the Hilbert class field of cyclotomic fields always were 1: see http://mathoverflow.net/questions/13428/please-check-my-6-line-proof-of-fermats-last-theorem/13469#13469 – Franz Lemmermeyer May 09 '15 at 15:04
  • @FranzLemmermeyer Your and David Loeffler's comments do help finding true motivation for the class tower problem. Thanks! –  May 10 '15 at 16:14
  • BTW, it is not even known that there are infinitely many fields with class number 1. – Ian Agol Jun 25 '17 at 04:37

1 Answers1

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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.

Olivier
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  • @MarkSapir No problem, I'm always happy to squash the hope of proving FLT by elementary means ;-). – Olivier Apr 23 '15 at 18:42
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    The question was more about motivation. I always thought that FLT was the motivation for the class tower problem. But it looks like there is no close relation. –  Apr 23 '15 at 20:15
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    @MarkSapir Hilbert care about class fields because it is easier to prove reciprocity laws at primes splitting completely (and that's why he introduced the class field as field in which principal ideals split completely, and not as an unramified extension as we do today). As I learned from Franz Lemmermeyer (which is the go to guy for these questions), from that it is rather natural to ask what is the class group of a Hilbert class field (because if the Hilbert class field is factorial, then all the better for the proof) and whence the class field tower problem. FLT has nothing to do with it. – Olivier Apr 23 '15 at 20:53
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    Thank you! It does make sense. But at least FLT was the main reason for defining the class number. Right? –  Apr 23 '15 at 21:08
  • @MarkSapir there is a question by Emerton, with an answers by Lemmermeyer and others, on the relative importance of FLT and reciprocity laws for Kummer's work that contains some information related to this. –  Apr 24 '15 at 00:57
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    @quid I looked at that discussion. As far as I understand from Lemmermeyer's paper (the link on that page is broken, but I found the paper anyway), it is more about ideals and their unique prime decompositions, i.e., class number 1, than about general class number. It is good to know that reciprocity laws played important role in the development of the theory of ideal numbers. I did not know that before. FLT also played a role, whether decisive or not - it is not that important to me. –  Apr 24 '15 at 12:48
  • @MarkSapir I agree the situation is not clear-cut, which is why I only mentioned the resource rather than answering your question directly. (I also replaced the link there; if you have a moment could you perhaps double check we found the same paper, so that I do not add a wrongish link.) –  Apr 24 '15 at 13:14
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    @quid Yes, it is the same paper, published in Abh. Math. Semin. Univ. Hambg. (2009) 79: 165–187. –  Apr 24 '15 at 13:19