# Infinite Galois correspondence “according to Artin”

Ever since Artin's lectures on Galois Theory one knows how to set up and derive the usual Galois correspondence in the finite(-dimensional) case using just a bit of elementary Linear Algebra, and without first introducing and employing the conditions of "normality" and "separability" of the finite field extension being considered. To be sure, if not only for applications, these conditions have to be considered later on, but the fact remains how elegantly the actual correspondence may be derived, by assuming the small field to be the fixed field corresponding to a finite subgroup of the automorphism group of the large field. By the introduction of the Krull- (or finite) topology the Galois correspondence may be stated and proven in the infinite algebraic case, as is well-known, but all presentations I know of either define infinite Galois extensions of fields as algebraic, normal and separable, and/or prove the main results using these conditions. For years I've been wondering whether there exist equally slick presentations of infinite Galois theory using the fact that the small field should correspond to a compact subgroup of the large field's automorphism group, without taking the way via normal and separable finite extensions, but directly using some general topological arguments to help replace "finiteness" by "compactness" ? Any help or (available) reference would be greatly appreciated ! Kind regards, Stephan.

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What references have you looked at for infinite Galois theory? For example, have you looked at Bastida's Field Extensions and Galois Theory? –  KConrad Apr 23 '11 at 10:34
Bastida's monograph is where I actually first encountered the term "finite topology", I believe, but seem to recall that the approach taken in the infinite case is not the one I am looking (or hoping) for. I believe he does prove some nice generalizations (Dedkind-Kaplansky) of "Artin's lemma", but the topology of the Galois group as such is not really exploited in the infinite case, and everything is (once again) played down to the finite case, using normality and separability. But thanks for the comment, anyway ! Kind regards, Stephan. –  Stephan F. Kroneck Apr 23 '11 at 16:07