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

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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 '11 at 11:47
Please don't post at two places simultaneously.… – Zev Chonoles Feb 13 '11 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 '11 at 8:12

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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