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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.

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    $\begingroup$ Maybe the much-fabled Lenstra lectures notes on Galois theory for schemes? (Theorem 1.11 seems to be what you want, except it is even a further generalization.) websites.math.leidenuniv.nl/algebra/GSchemes.pdf $\endgroup$
    – R.P.
    Commented Oct 21 at 18:48
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    $\begingroup$ @R.P. Are you sure that this one is proved without using Galois theory? Thm 2.9 loc. cit. is the version for fields, and its proof seems to use Galois theory, if I understand correctly. $\endgroup$
    – Z. M
    Commented Oct 21 at 20:15
  • $\begingroup$ @Z.M I am not completely sure, but I had the impression that the field version is proved just in order to provide a motivating example, and the proof of the general result does not depend on the proof of the field version. But once again, not sure. $\endgroup$
    – R.P.
    Commented Oct 21 at 20:21
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    $\begingroup$ Have you looked at Dress's paper One more short cut to Galois theory sciencedirect.com/science/article/pii/S0001870885710055. He proves this statement using, essentially, Galois descent and then proves the fundamental theorem of Galois theory using this approach. $\endgroup$
    – Benjamin Steinberg
    Commented Oct 23 at 15:28
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    $\begingroup$ I think also the above theorem is proved directly in Algebre Theories Galoisiennes by Douady and Douady if I recall correctly $\endgroup$
    – Benjamin Steinberg
    Commented Oct 23 at 15:29

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