Galoisian Galois Theory

TL;DR

This paper offers an exposition of Galois Theory emphasizing the connection between Lagrange's combinatorial approach and Galois algebraic extensions, with a focus on algorithmic methods suited for computational applications.

Contribution

It bridges classical Galois Theory with modern computational approaches, prioritizing algorithmic techniques over traditional simplified presentations.

Findings

Highlights the connection between Lagrange's approach and Galois extensions

Develops algorithmic methods for Galois Theory

Provides a computational perspective on Galois constructions

Abstract

These notes are an exposition of Galois Theory from the original Lagrangian and Galoisian point of view. A particular effort was made here to better understand the connection between Lagrange's purely combinatorial approach and Galois algebraic extensions of the latter. Moreover, stimulated by the necessities of present day computer explorations, the algorithmic approach has been given priority here over every other aspect of presentation. In particular, you may not find here the clean simplistic look characteristic of the classical exposition of E. Artin. In contrast these notes should provide a good starting point in attempting constructions in this most difficult computational arena.