7
$\begingroup$

Does the Galois group of every infinite $p$-extension $K$ of a number field $k$ contain a (closed) subgroup such that the quotient group is isomorphic to $\mathbb{Z}_p$?

My feeling is "yes", but I'm not sure if we need the extension $K/k$ to be abelian. My intuition would be that the "infinite" part of $Gal(K/k)$ comes from at least one copy of $\mathbb{Z}_p$. Am I right about this?

Thank you in advance!

Update (11/21/2013): In response to hunter's answer, I want to ask, under which conditions does a $p$-extension contain the $\Bbb Z_p$-extension? Update (28/11/2013): The question above is possibly too broad and not answerable.

$\endgroup$
1
  • $\begingroup$ @BrunoJoyal Sorry, I was mixing two different things. You are, of course, right, that the finiteness of the class number $h_k$ implies that the extension of the Hilbert class field of $k$ is finite and therefore every quotient as well. I removed the background. $\endgroup$
    – BIS HD
    Nov 18 '13 at 13:55
8
+50
$\begingroup$

No, take the compositum of all quadratic extensions of $\mathbb{Q}$.Every element of the Galois group is $2$-torsion, so this cannot possibly have a copy of $\mathbb{Z}_2$ as a quotient. If you allow only finitely many primes to ramify, then the answer becomes yes (because your Galois group is then finitely topologically generated, and now appeal to the structure theory of $p$-groups).

$\endgroup$
4
  • $\begingroup$ Thank you! So if we restrict ourselves to extensions $K/k$ where $K \subset k_S$ is in the maximal unramified $p$-extension outside $S$ for a finite set of primes $S$ of $k$, then - if $K/k$ is infinite, it contains $\mathbb{Z}_p$? Have you got a reference for the last statement concerning the structure theory of $p$-groups?) $\endgroup$
    – BIS HD
    Nov 18 '13 at 14:37
  • $\begingroup$ Now that I think about it, we need more than just group theory, since an infinite pro $p$ group can have finite abelianization. I retract my broader claim! $\endgroup$
    – hunter
    Nov 19 '13 at 10:51
  • $\begingroup$ Could you be more detailed about your thoughts of a pro $p$ group having finite abelianization? Thanks :-) $\endgroup$
    – BIS HD
    Nov 20 '13 at 10:43
  • 1
    $\begingroup$ When I made my claim "the answer becomes yes" I was wrongly thinking that the abelianization of an infinite pro $p$ group is still infinite, and then the claim is just that an infinite, finitely generated $\mathbb{Z}_p$ module has $\mathbb{Z}_p$ as a quotient. However, that claim was wrong (I googled to find explicit counterexamples). $\endgroup$
    – hunter
    Nov 20 '13 at 11:14

Your Answer

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.