Does anyone know the earliest use of the term "Galois extension"? I thought it might be in Emil Artin's Notre Dame lectures but I couldn't find it there. (He does use the terms "normal" and "separable.")
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$\begingroup$ The earliest occurrence in MathSciNet seems to be in the review of "Une généralisation de la notion de corps—corpoïde. Un corpoïde remarquable de la théorie des corps valués" by Marc Krasner, C. R. Acad. Sci. Paris 219, (1944) 345–347. The relevant sentence is "Commutative extension corpoids are discussed and Galois extensions defined." $\endgroup$– Timothy ChowCommented Apr 11, 2019 at 2:49
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Not an answer, but an extended comment. Probably the terminology is not due to Artin. The Notre Dame lectures were in 1942 and published in 1944. In Emil Artin's 1947 lecture notes (notes taken by Albert A Blank, and seems to have been printed by Courant Institute; I'm not sure as I only have a photocopy) what normally is now called a Galois extension he calls a "normal extension" (he used normal differently from how it is used today).
This is also corroborated by a footnote in Hungerford's Algebra. In my edition the definition of a Galois extension (Definition 2.4 in Chapter V) has a footnote that reads (emphasis mine):
A Galois extension is frequently required to be finite dimensional or at least algebraic ... equivalent to the usual one. Our definition is essentially due to Artin, except that he calls such an extension "normal." Since this use of "normal" conflicts (in case char $F\neq 0$) with the definition of "normal" used by many other authors, we have chosen to follow Artin's basic approach, but to retain the (more or less) conventional terminology.