Purely Inseparable Galois theory I: The Fundamental Theorem

TL;DR

This paper develops a Galois correspondence for finite purely inseparable field extensions, extending classical results to more general cases beyond exponent one.

Contribution

It generalizes Jacobson's classical Galois correspondence to finite purely inseparable extensions of arbitrary exponent.

Findings

Established a Galois correspondence for purely inseparable extensions

Extended classical Galois theory to new classes of field extensions

Provided a framework for understanding automorphisms in inseparable cases

Abstract

We construct a Galois correspondence for finite purely inseparable field extensions $F/K$, generalising a classical result of Jacobson for extensions of exponent one (where $x^p \in K$ for all $x\in F$).