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

closed as too localized by S. Carnahan♦ Feb 14 2011 at 8:11

1 Answer

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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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