I have proved for myself the following theorem, generalizing Galois theorem to general algebraic extensions. My question is: is it true, and is there some reference to this theorem in the literature?

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 $H\mapsto M = {\rm Fix}(H)$.


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