# Closure of Normal Subgroups of the Galois Group for an Infinite Galois Extension is Normal

Let K/F be an infinite Galois extension, and let N be a normal sub-group of Gal(K/F). Show that N closure is a normal subgroup of Gal(K/F).

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the closure of a normal subgroup in any topological group is normal –  user8268 Oct 31 '12 at 15:23
I have this proof. But can we do it w/o topological group? –  user41481 Oct 31 '12 at 15:30
Without topology, what do you mean by closure? –  franz lemmermeyer Nov 2 '12 at 13:56
I am seeking a proof using Fundamental Theorem of Infinite Galois Theory. Possibly by showing Fixed field of N closure is Galois over F. –  user41481 Nov 2 '12 at 14:57
Okay!! I have got it. Thanks, anyway! –  user41481 Nov 6 '12 at 11:27