Analogue of the Galois Theory for arbitrary finite field extensions

TL;DR

This paper extends classical Galois Theory to arbitrary finite field extensions by developing a ring-theoretic framework based on maximal symmetry principles, describing intermediate subfields via invariants of associated algebraic objects.

Contribution

It introduces a novel ring-theoretic approach to Galois Theory for all finite field extensions, generalizing existing theories beyond Galois extensions.

Findings

Established a correspondence between subfields and invariants of algebraic objects.

Characterized normal extensions via maximal symmetry conditions.

Extended Galois correspondence to arbitrary finite extensions.

Abstract

This paper is a finishing touch to the (over 200 years) {\em classical} `Galois Theory' of {\em arbitrary} finite field extensions, i.e. the goal of it is to describe intermediate subfields of an arbitrary finite field extension via {\em invariants} of `natural/obvious' objects that are associated with subfields via two Galois-type correspondences. The classical Galois Theory covers the case of finite Galois field extensions. For finite Galois field extensions the objects are their Galois groups and their invariants. In \cite{GaloisTh-RingThAp}, we introduce a new (ring theoretic) approach to the Galois Theory which is based on the {\em principle of maximal symmetry}. In \cite{AnGaloisTh-NORMAL-Fields}, the maximal symmetry of {\em normal} finite field extensions yields an analogue of the Galois Theory for them. For a normal finite field extension $L/K$ the `natural/obvious' objects are the subalgebra $\CD (L/K)\rtimes G(L/K)$ of $\End (L/K)$ that is generated by the automorphism group $G(L/K)$ and the algebra $\CD (L/K)$ of differential operators on $L/K$ and its `invariants'. The `maximal symmetry' means the equality $\End (L/K)=\CD (L/K)\rtimes G(L/K)$ which turns out to be a characteristic property of {\em normal} finite field extensions, \cite{AnGaloisTh-NORMAL-Fields}. The aim of this paper is to obtain an analogue of the Galois Theory for {\em arbitrary} finite field extensions based on results and ideas of \cite{GaloisTh-RingThAp} and \cite{AnGaloisTh-NORMAL-Fields}.