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Theorem: Recall that a subfield ##M## of a field ##L## is a perfect closure in ##L## if there is no purely inseparable extension of ##M## inside ##L##. In other words, ##\text{char}(M) = 0## or ##\text{char}(M) = p > 0## and all the ##p##-roots of elements of ##M## contained in ##L## already belong to ##M##.

Assume that ##L/K## is a normal extension of fields. Suppose this extension finite for the sake of simplicity (otherwise, consider only closed groups of automorphisms for the Krull topology). Galois theorem becomes:

The application ##M\mapsto H = {\rm Aut}(L/M)## define a ##1\!-\!1## correspondence, reversing the inclusion, between the perfect closures ##M## in ##L## between ##K## and ##L##, and the subgroups ##H## of ##\text{Aut}(L/K)##. The invert is given as usual by ##H\mapsto M = {\rm Fix}(H)##.