Abstract:
Avariste Galois gave a proof that there was no general solution by radicals to polynomial equations of degree five or higher. To do so, he established what is today known as the Galois correspondence, allowing him to apply methods from group theory to problems involving polynomial fields. I will give a brief overview of Galois\' proof, in terms of the polynomials he worked with (polynomoials over subfields of the complex field). I will also show how the theory can be used to produce an insoluble quintic.

Contents:

1. Historical Background
1.1 History of Polynomial Equations
1.2 Galois - Prodigious Beginnings
1.3 Galois' Contribution

2. Theorems and Definitions
2.1 Field Theory
2.2 Factorization of Polynomials
2.3 Field Extensions
2.4 Degree of an Extension
2.5 Normality and Separability
2.6 Group Theory
2.7 Fundamental Theorem of Galois Theory

3 Galois' Proff
3.1 Polynomials Solvable by Radicals
3.2 The Galois Correspondence
3.3 Group Theory
3.3.1 Normality
3.3.2 Separability
3.3.3 Soluable Groups
3.4 A Quintic Not Solvable By Radicals