Parametric Polynomials for Small Galois Groups

Author (Your Name)

Claire HuangFollow

Date of Award

2018

Document Type

Honors Thesis (Open Access)

Department

Colby College. Mathematics and Statistics Dept.

Advisor(s)

Fernando Q. Gouvea

Second Advisor

David Krumm

Abstract

Galois theory, named after French mathematician Evariste Galois in 19th-century, is an important part of abstract algebra. It brings together many different branches of mathematics by providing connections among fields, polynomials, and groups.

Specifically, Galois theory allows us to attach a finite field extension with a finite group. We call such a group the Galois group of the finite field extension. A typical way to attain a finite field extension to compute the splitting field of some polynomial. So we can always start with a polynomial and find the finite group associate to the field extension on its splitting field. This focus is well-understood, but the converse remains puzzling to people. The Inverse Galois Problem asks the mysterious question of whether for any finite group, there exists a finite field extension that is associate to it. Furthurmore, is there a polynomial whose splitting field that can make such a field extension? This mystry is part of the motivation for my thesis. The other motivation is the desire to repace this existence problem with constructive problem. We will write out some families of polynomials whose splitting fields over the rational field have certain small groups as their Galois group.

Keywords

Galois Theory, Inverse Galois Problem

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