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arXiv · 2511.21353

Analogue of the Galois Theory for arbitrary finite field extensions

V. V. Bavula
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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}.

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V. V. Bavula. 2025-11-26. Analogue of the Galois Theory for arbitrary finite field extensions. https://arxiv.org/abs/2511.21353

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