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It is known that if K is a finite Galois extension of Q with Galois group G, then G is generated by the inertia groups of ramified primes in the extension.

Does the statement hold for infinite Galois extensions?

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    $\begingroup$ Doesn't it have to do with the fact that $\mathbf{Q}$ has no unramified extension of degree $>1$ ? $\endgroup$ Feb 13, 2011 at 11:47
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    $\begingroup$ Please don't post at two places simultaneously. math.stackexchange.com/questions/21829/… $\endgroup$ Feb 13, 2011 at 17:43
  • $\begingroup$ If you want this question to be reopened, please delete the cross-posted question and flag for moderator attention. $\endgroup$
    – S. Carnahan
    Feb 14, 2011 at 8:12

1 Answer 1

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Writing an infinite Galois group as the projective limit of finite Galois groups, one sees that the inertia groups in the infinite extension topologically generate the infinite Galois group.

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