A simple proof of the fundamental theorem of Galois theory

TL;DR

This paper provides a straightforward proof of the fundamental theorem of Galois theory, linking intermediate fields and Galois subgroups, based on a combinatorial property of fields.

Contribution

It introduces a simple proof of the fundamental theorem of Galois theory using a combinatorial argument about fields and subfields.

Findings

Establishes a correspondence between intermediate fields and Galois subgroups.

Uses a combinatorial fact that a field cannot be the union of finitely many proper subfields.

Abstract

We present a simple proof of the fundamental theorem of Galois theory, which establishes a correspondence between the intermediate fields of a finite Galois extension and the subgroups of its Galois group. The proof is based on the combinatorial fact that a field cannot be expressed as the union of finitely many proper subfields.