In the generalized fundamental theorem of Galois theory, theorem says there is a one-to-one correspondence between the set of all intermediate fields of extension and the set of all $\textbf{closed}$ subgroups of the Galois group $Aut_KF$. It seems to say that there can be some subgroup of the Galois group that is not closed. Is it true? If so, please give me an example. Thanks in advance.

  • $\begingroup$ It seems to be you're thinking of profinite groups, but then a counterexample may be hard to come up with...look here for some ideas (read below the abstract) about normal non-closed subgroups: ma.rhul.ac.uk/profinite_groups/Notes_Nikolay_Nikolov.pdf $\endgroup$ – DonAntonio Feb 15 '14 at 12:02
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    $\begingroup$ The Galois group of a finite field $\mathbb{F}_p$ is the profinite group $\hat{\mathbb{Z}}$, and $\mathbb{Z}$ is a non-closed subgroup of $\hat{\mathbb{Z}}$. $\endgroup$ – Zhen Lin Feb 15 '14 at 12:13

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