It is well-known that Galois correspondence sends a normal subgroup to a normal extension of a field. Specifically, given a Galois extension $L/K$ and the corresponding Galois group $G$, normal subgroups of $G$ correspond to normal subextensions $F/K$ .

Is there a characterization of the subextensions corresponding to characteristic subgroups?


That question doesn't make sense because automorphisms of $Gal(L/K)$ don't have any meaning in Galois theory.

Only the inner automorphisms have a meaning, and a subgroup sent to itself by every inner automorphism is called a normal subgroup.


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