# The development of Galois theory

### Fiona Brunk

### Introduction

Modern algebra may be described as the study of the structure of sets with certain properties and their interaction with one another via specific operations. However, merely 200 years ago, algebraists were primarily, if not exclusively, concerned with solving equations. The evolution of algebra from what is sometimes called "school algebra" today to the study of abstract structures largely coincides with the development of Galois Theory in the 19

In 1971, B. Kiernan published an article [3] on

To put Galois' work into perspective, consider the historical background of the mathematical community to which Galois presented his ideas. During his lifetime, algebraists were interested in the solvability of equations of degree higher than four. The formula for the solution of the quadratic equation had essentially been known to the Babylonians and the general equations of degree three and four were solved during the Renaissance in Italy. Yet no one managed to find a general solution of the quintic and in the early 1800s, Ruffini and Abel proved that this was impossible. However, this general result has no consequences for the algebraic solvability of any specific equation of degree five, and algebraists were seeking criteria which would determine whether a given quintic was solvable by radicals.

It was in this context that Galois published his work. Today, we see Galois Theory as the study of the structure of fields and their automorphism groups and although Galois used different terminology, this was essentially how he conceived his theory. As one possible application, he described how his theory could be used to determine necessary and sufficient conditions for the solvability of an equation by radicals. Although Galois pointed out that he was presenting "the general principles and just one application" [2] of his theory, it is not surprising, given the historical context, that the mathematicians of his time concentrated on this application and largely ignored the general theory. They were trying to extract from Galois' work a procedure for finding the solutions of any given special equation, although Galois himself had never intended for his theory to yield such a procedure. It was only much later in the 19

^{th}and early 20^{th}century. The concept of a group is generally credited to the French mathematician Évariste Galois, and while the idea of a field was developed by German mathematicians such as Kronecker and Dedekind, Galois Theory is what connects these two central concepts in algebra, the group and the field.In 1971, B. Kiernan published an article [3] on

*The Development of Galois Theory from Lagrange to Artin*. While this article is a very well researched and comprehensive study, it runs over 114 pages and so a summary of it would be useful to have as a first point of reference. This project attempts to give such a summary and will thus contain very few original points, but instead try to summarise Kiernan's main points.To put Galois' work into perspective, consider the historical background of the mathematical community to which Galois presented his ideas. During his lifetime, algebraists were interested in the solvability of equations of degree higher than four. The formula for the solution of the quadratic equation had essentially been known to the Babylonians and the general equations of degree three and four were solved during the Renaissance in Italy. Yet no one managed to find a general solution of the quintic and in the early 1800s, Ruffini and Abel proved that this was impossible. However, this general result has no consequences for the algebraic solvability of any specific equation of degree five, and algebraists were seeking criteria which would determine whether a given quintic was solvable by radicals.

It was in this context that Galois published his work. Today, we see Galois Theory as the study of the structure of fields and their automorphism groups and although Galois used different terminology, this was essentially how he conceived his theory. As one possible application, he described how his theory could be used to determine necessary and sufficient conditions for the solvability of an equation by radicals. Although Galois pointed out that he was presenting "the general principles and just one application" [2] of his theory, it is not surprising, given the historical context, that the mathematicians of his time concentrated on this application and largely ignored the general theory. They were trying to extract from Galois' work a procedure for finding the solutions of any given special equation, although Galois himself had never intended for his theory to yield such a procedure. It was only much later in the 19

^{th}century that mathematicians became interested in Galois Theory for its own sake, beyond its application to the solvability of equations. This development of Galois Theory will now be looked at in some greater detail.