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Abstract

We give a short and self-contained proof of the Fundamental Theorem of Galois Theory (FTGT) for finite degree extensions. We derive the FTGT (for finite degree extensions) from two statements, denoted (a) and (b). These two statements, and the way they are proved here, go back at least to Emil Artin (precise references are given below). The argument is essentially taken from Chapter II of Emil Artin’s Notre Dame Lectures [A]. More precisely, statement (a) below is implicitly contained in the proof Theorem 10 page 31 of [A], in which the uniqueness up to isomorphism of the splitting field of a polynomial is verified. Artin’s proof shows in fact that, when the roots of the polynomial are distinct, the number of automorphisms of the splitting extension coincides with the degree of the extension. Statement (b) below is proved as Theorem 14 page 42 of [A]. The proof given here (using Artin’s argument) was written with Keith Conrad’s help. Theorem Let E/F be an extension of fields, let a1, . . . , an be distinct generators of E/F such that p := (X − a1) · · · (X − an) is in F [X]. Then • the group G of automorphisms of E/F is finite, • there is a bijective correspondence between the sub-extensions S/F of E/F and the subgroups H of G, and we have S ↔ H ⇐⇒ H = Aut(E/S) ⇐⇒ S = E =⇒ [E : S] = |H|, where E is the fixed subfield of H, where [E : S] is the degree (that is the dimension) of E over S, and where |H| is the order of H.