The Galois Theory (a ring theoretic approach)

Abstract

The fundamental concepts in the Galois Theory are separable, normal and Galois field extensions. These concepts are central in proofs of the Galois Theory. In the paper, we introduce a new approach, a ring theoretic approach, to the Galois Theory which is based on central simple algebras and none of the above concepts are used or even mentioned. The only concept which is used is `G-extension' (a finite field extension $L/K$ is called a {\em G-extension} if the endomorphism algebra ${\rm End}_K(L)$ is generated by the field $L$ and the automorphism group ${\rm Aut}_{K-{\rm alg}}(L)$). So, G-extensions are the most symmetric field extensions. In this approach, the Galois Correspondences (for subfields and Galois subfields of $L$) are deduced from the Double Centralizer Theorem which is applied to the central simple algebra ${\rm End}_K(L)$ and G-extensions. Since the class of G-extensions {\em coincides} with the class of Galois extensions, all main results of the Galois Theory are obtained from the `corresponding' results for G-extensions. This approach gives a new conceptual (short) proofs of key results of the Galois Theory. It also reveals that the `maximal symmetry' (of field extensions) is the essence of the Galois Theory.