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?
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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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. 

