ABSTRACT
Galois theory is a beautiful area of mathematics that provides connections between polynomials, fields, and groups. One of the fundamental applications of Galois theory is the resolution of the question of which polynomials can be solved by radicals. Galois theory is built on characterizing solvable polynomial equations in terms of field extensions. By proving that the symmetric group of degree 5 or higher is not solvable, the authors can demonstrate the existence of polynomials whose Galois groups are these symmetric groups and are therefore not solvable by radicals. These observations are the key components to Galois theory, and they devote much of this investigation to addressing them. They will see how to use the Galois correspondence to translate a problem of polynomials and fields to a problem in group theory.
