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Hello!

Let $K$ be a number field. All abelian unramified extensions are contained in the Hilbert class field which is a finite extension 'maximal' with respect to this property. For general unramified extensions, is there a bound (depending on $K$) on the degree of an unramified extension over $K$? If so, does the compositum of all unramified extensions also have finite degree over $K$ in general?

Thanks for your attention!

ADDENDUM: as Hunter noticed the answer can be no even just even for solvable groups, when the field admits an infinite class field tower. But perhaps it is still interesting to study the question for extension having simple Galois group, and possibly their compositum. Is there anything known about this case?

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    $\begingroup$ Yes, much is known about the case in your addendum as well. I recommend trying to decide exactly what you want to know and re-asking as a new question. You might also try clicking around the "Related" links on the right-hand side -- there is much information therein. $\endgroup$
    – Cam McLeman
    Commented Jan 28, 2011 at 16:00
  • $\begingroup$ Related: mathoverflow.net/questions/76616 $\endgroup$
    – Watson
    Commented Jul 5, 2018 at 20:53
  • $\begingroup$ Related: mathoverflow.net/questions/195767 $\endgroup$
    – Watson
    Commented Jul 6, 2018 at 9:08

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No- even the process of iteratively taking the Hilbert class field, the Hilbert class field of the Hilbert class field, etc, need not terminate. See

https://en.wikipedia.org/wiki/Golod-Shafarevich_theorem

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  • $\begingroup$ Right... thanks for your anwer. But perhaps the question can be refined: what about extension containg no proper abelian extension? is anything know about this case? $\endgroup$
    – Maurizio Monge
    Commented Jan 28, 2011 at 15:02
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In addition to Hunter Brooks' answer let me mention that C. Maire has constructed number fields with class number 1 that admit an infinite unramified extension. See also D. Brink, Remark on infinite unramified extensions of number fields with class number one, J. Number Theory 130 (2010), 304-306 for a recent modification of Maire's idea.

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    $\begingroup$ Interesting, thanks, that clearly provides an unramified field extensions whose maximal abelian subextension is trivial. But if i understood correctly the paper provides a field $K$ with classes number $1$, and simple non-abelian unramified extension $L/K$ that has an infinite class field tower, and we have that the Galois group of any finite subextension is the extension of a simple group by a solvable group, and in particular there is just one non-abelian simple quotient. So the question of the existence of infinite unimified simple estensions is still open. $\endgroup$
    – Maurizio Monge
    Commented Jan 28, 2011 at 17:03
  • $\begingroup$ See mathoverflow.net/questions/53712/infinite-simple-galois-groups $\endgroup$
    – Franz Lemmermeyer
    Commented Jan 30, 2011 at 19:10
  • $\begingroup$ Reposting the link mentioned in the previous comment so that it appears in the "Linked" questions list: Infinite simple Galois groups $\endgroup$
    – The Amplitwist
    Commented Mar 24, 2023 at 11:57
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It is worth pointing out a recent paper by Manabu Ozaki in the Inventiones where he proves that

Given any finite $p$-group $G$ ($p$ being a prime number), there exists a number field $F$ such that the group of $F$-automorphisms of the maximal unramified $p$-extension of $F$ is isomorphic to $G$.

See his Theorem I. The paper is also available at arXiv:0705.2293 [math.NT].

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