An alternative approach to the concept of separability in Galois theory

TL;DR

This paper introduces a new perspective on separability in Galois theory, providing alternative proofs for fundamental results without relying on minimal polynomials or derivatives.

Contribution

It offers an innovative approach to separability, simplifying proofs of key theorems in Galois theory through a different conceptual framework.

Findings

New proofs of the existence of the separable closure

Alternative proof of the primitive element theorem

Demonstration of transitivity of separability

Abstract

The notion of a separable extension is an important concept in Galois theory. Traditionally, this concept is introduced using the minimal polynomial and the formal derivative. In this work, we present an alternative approach to this classical concept.Based on our approach, we will give new proofs of some basic results about separable extensions (such as the existence of the separable closure, Theorem of the primitive element and the transitivity of separability).