Introduction

Figure 1.1: A portrait of Évariste Galois at age 15. He died at the young age of 20 in a duel fight. Despite this, he had left his impact on mathematics forever.

The Galois group is one of the derivatives of Galois theory and has been utilized to solve many long standing questions in mathematics. For centuries, mathematicians have wondered if it is possible to know whether polynomials of degree 5 or higher are all solvable, or whether certain geometries can be constructed using an unmarked straightedge and a compass. Mathematicians had found ways to answer some of these questions using other various methods, but the the discovery of Galois groups provided an elegant way to prove some of these facts. Galois theory and Galois groups were discovered by Évariste Galois, for which Galois groups are named. In the following centuries, mathematicians have expanded upon the work of Galois after his early death at 20. Although Galois groups had been defined in ways of varying rigor throughout history and expanded upon in different ways, each aspect of Galois group will be described in its modern notation.

The Galois group and Galois theory are aspects of abstract algebra that are often discussed at the graduate level and sometimes in an undergraduate abstract algebra course. This article is dedicated to the curious undergraduate student of a mathematical background of varying rigor, so the concepts described in this article are presented to give the reader an introduction and overview of the topic. In order to fully appreciate and understand the Galois group, an understanding of groups and fields is an absolute necessity for any sort of meaningful comprehension of what the Galois group truly is. Three major ideas leading up to Galois groups are the ideas of groups, fields and field extensions.

Groups - A set of numbers that must fulfill four conditions for an operation

Fields - A set of numbers that can experience the typical operations of addition, multiplication, division, and subtraction

Field Extension- An extension of a already existing field to create a new "super field"

These will be described further in depth in the following sections. Before one proceeds, it is assumed that one has an understanding of basic number systems, the real numbers, the rational numbers, along with many other common number systems. The idea of a set is referred to multiple times to describe various concepts that make use of them.


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