Let $G_\Bbb Q$ be the absolute Galois group of $\Bbb Q$, class field theory describes the structure of $G_\Bbb Q^{ab}$. But I feel something strange about the definition of $G_\Bbb Q^{ab}$: Is $[G_\Bbb Q,G_\Bbb Q]$ a closed subgroup of $G_\Bbb Q$?
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As mentioned in Field and Galois Theory by P. Morandi, we have $\mathrm{Gal}(\Bbb Q^{\mathrm{ab}} / \Bbb Q) \cong G_{\Bbb Q} / \overline{ [G_{\Bbb Q}, G_{\Bbb Q}] }$. In general, the quotient of a Hausdorff group by a non-closed subgroup might not be Hausdorff (see $\Bbb R / \Bbb Q$), while any profinite group is Hausdorff. – Watson Feb 16 '18 at 16:43
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This is true if we replace $\Bbb Q$ by $\Bbb Q_p$ (see this comment). – Watson Jul 09 '18 at 14:59
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Indeed, it is not true in general, that the commutator subgroup $[G,G]$ of a profinite group $G$ is a closed subgroup:
Is the commutator subgroup of a profinite group closed?
So the quotient $$ G_K/[G_K,G_K] $$ is by the closure of the (usual algebraic) commutator subgroup of $G_K$, the smallest subgroup which gives an abelian quotient.
Dietrich Burde
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