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?
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.