In https://www.jmilne.org/math/CourseNotes/FT.pdf Milne's Galois theory notes- chapter 8, he remarks that it is possible to classify etale algebras without using Galois theory then deduce Galois theory and he will explain this sometime. Does any one know the reference for the same (either by Milne or someone else's material)?

In Milne's Galois theory notes — chapter 8, quoted below, he remarks that it is possible to classify étale algebras without using Galois theory then deduce Galois theory and he will explain this sometime. Does any one know the reference for the same (either by Milne or someone else's material)?

THEOREM 8.21 The functor $A \leadsto \mathscr F(A)$ is a contravariant equivalence from the category of étale $F$-algebras to the category of finite $G$-sets with quasi-inverse $\mathscr A$.

PROOF. This summarizes the results in the last three propositions. ▢

It is possibe to prove Theorem 8.21 directly, without using Galois theory, and then deduce Galois theory from it. Perhaps I'll explain this sometime.