ABSTRACT
28.1 Yes, Galois Theory is really as important as it looks! If not for its actual contents, then certainly for the innovative quality of ideas. We have tried to highlight this importance in the historical comments. A normal field (extension) is defined as a splitting field of some polynomial. and we establish that every field has a normal closure (Proposition 28.17). For normal fields there is a better connection between the algebraic structure of the field and automorphisms of it (e.g. Theorem 28.23). This connection is most tight when the field is both normal and separable. We define (finite) Galois extensions in 28.49 as a finite extension E/F such that F is the field of invariants of some group of automorphisms of E, then we prove in Theorem 28.50 that finite Galois extensions are exactly normal and separable extensions. This approach has the advantage that it reduces the basic ingredients of the theory to a combination of Dedekind's Lemma (Theorem 28.25) and E. Artin's Theorem (Theorem 28.48) One look at these results will convince the reader that Galois Theory is now essentially linear algebra e.g. the proofs of the results mentioned depend only on the solution of linear equations.
