Analogue of the Galois Theory for arbitrary finite field extensions
V. V. Bavula
TL;DR
The paper addresses extending Galois theory to arbitrary finite field extensions by formulating two Galois-type correspondences based on the endomorphism algebra $E(L/K)$ and the differential-operator algebra ${ m D}(L/K)$. It develops a unified framework for subfield and normal-subfield correspondences, with separate treatments for separable and purely inseparable cases, and shows how normal extensions recover the classical theory while purely inseparable ones are governed by differential operators. Key contributions include explicit bijections between subfields and appropriate subalgebras (via invariants like $L^{{ m D}(L/K)_+}$ and $L^{{G(L/K) ext{}}}$), plus concrete results for B-extensions and normal extensions, including a detailed normal-extension example. The work provides a characteristic-free, ring-theoretic perspective that clarifies the role of maximal symmetry in Galois Theory and offers practical descriptions of intermediate fields through operator- and group-invariants.
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}.