12
$\begingroup$

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)?

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – David Loeffler Apr 23 '15 at 7:09
  • $\begingroup$ @DavidLoeffler: Would the case when $p$ is prime imply FLT? $\endgroup$ – user6976 Apr 23 '15 at 7:15
  • $\begingroup$ Yes, because anything implies a true statement :-). Seriously, why should this question have any particular relation to FLT? $\endgroup$ – David Loeffler Apr 23 '15 at 7:18
  • $\begingroup$ @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? $\endgroup$ – user6976 Apr 23 '15 at 7:21
  • 1
    $\begingroup$ 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. $\endgroup$ – David Loeffler Apr 23 '15 at 11:26
11
$\begingroup$

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.

| cite | improve this answer | |
$\endgroup$
  • 1
    $\begingroup$ 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. $\endgroup$ – user6976 Apr 23 '15 at 20:15
  • 1
    $\begingroup$ @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. $\endgroup$ – Olivier Apr 23 '15 at 20:53
  • 1
    $\begingroup$ Thank you! It does make sense. But at least FLT was the main reason for defining the class number. Right? $\endgroup$ – user6976 Apr 23 '15 at 21:08
  • 1
    $\begingroup$ @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. $\endgroup$ – user6976 Apr 24 '15 at 12:48
  • 1
    $\begingroup$ @quid Yes, it is the same paper, published in Abh. Math. Semin. Univ. Hambg. (2009) 79: 165–187. $\endgroup$ – user6976 Apr 24 '15 at 13:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy