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Analogue of the Galois Theory for normal fields and B-extensions (characteristic free approach)

V. V. Bavula

Abstract

The aim of the paper is to introduce B-extensions which are the most symmetrical finite field extensions (a finite field extension $L/K$ is called a {\it B-extension} if the endomorphism algebra ${\rm End}_K(L)$ is generated by the algebra of differential operators ${\cal D} (L/K)$ on the $K$-algebra $L$ and the automorphism group $G(L/K):={\rm Aut}_{K-{\rm alg}}(L)$) and to obtain an analogue of the Galois Theory for B-extensions. Surprisingly, the class of B-extensions coincides with the class of {\it normal } finite field extensions. As a result, an analogue of the Galois Theory is obtained for normal field extensions. In particular, all Galois field extensions and all purely inseparable field extensions are B-extensions. Our approach is a ring theoretic (characteristic free) approach which is based on central simple algebras. In this approach, analogues of the Galois Correspondences (for subfields and normal subfields of $L$) are deduced from the Double Centralizer Theorem which is applied to the central simple algebra ${\rm End}_K(L)$ and subfields of B-extensions. Since Galois finite field extensions are B-extensions, this approach gives a new conceptual (short) proofs of key results of the Galois Theory, see [2] for details. It also reveals that the `maximal symmetry' (of field extensions) is the essence of the classical Galois Theory and the analogue of the Galois Theory for normal field extensions.

Analogue of the Galois Theory for normal fields and B-extensions (characteristic free approach)

Abstract

The aim of the paper is to introduce B-extensions which are the most symmetrical finite field extensions (a finite field extension is called a {\it B-extension} if the endomorphism algebra is generated by the algebra of differential operators on the -algebra and the automorphism group ) and to obtain an analogue of the Galois Theory for B-extensions. Surprisingly, the class of B-extensions coincides with the class of {\it normal } finite field extensions. As a result, an analogue of the Galois Theory is obtained for normal field extensions. In particular, all Galois field extensions and all purely inseparable field extensions are B-extensions. Our approach is a ring theoretic (characteristic free) approach which is based on central simple algebras. In this approach, analogues of the Galois Correspondences (for subfields and normal subfields of ) are deduced from the Double Centralizer Theorem which is applied to the central simple algebra and subfields of B-extensions. Since Galois finite field extensions are B-extensions, this approach gives a new conceptual (short) proofs of key results of the Galois Theory, see [2] for details. It also reveals that the `maximal symmetry' (of field extensions) is the essence of the classical Galois Theory and the analogue of the Galois Theory for normal field extensions.

Paper Structure

This paper contains 8 sections, 68 theorems, 209 equations.

Table of Contents

  1. Introduction
  2. Description of subalgebras of the endomorphism algebra $E(L/K)$ that contain the field $L$
  3. Algebras of differential operators on finite field extensions
  4. Analogue of the Galois correspondence for purely inseparable field extensions
  5. The algebras ${\cal D} (L/K)\rtimes G(L/K)$, $L\rtimes G(L/K)$ and ${\cal D} (L/K)$
  6. The classes of B-extensions and normal finite field extensions coincide
  7. Analogue of the Galois Theory for normal field extensions
  8. Field extensions $L/L^{G(L/K)}_{dif}$ and their properties

Key Result

Theorem 2.1

(The Noether-Skolem Theorem) Let ${\cal A}\in {\mathfrak{C}}(K)$, $A$ and $B$ be isomorphic, simple subalgebras of ${\cal A}$. Then $B=uAu^{-1}$ for some unit $u\in {\cal A}^\times$.

Theorems & Definitions (147)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • Definition 2.4
  • Theorem 2.5
  • ...and 137 more