On the Galois groups of sextic trinomials

• Main Author: Brown, Stephen Christopher
• Language: English
• Published: University of British Columbia 2011
• Online Access: http://hdl.handle.net/2429/36998

It is well known that the general polynomial a_n*x^n + a_{n-1}*x^{n-₋¹} + ... + a₁x + a_0 cannot be solved algebraically for n ≥ 5; that is, it cannot be solved in terms of a finite number of arithmetic operations and radicals. We can, however, associate every irreducible sextic polynomial with a Galois group. The Galois group of a given polynomial can give us a great deal of information about the nature of the roots of a polynomial and it can also tell us if the polynomial itself is algebraically solvable. This leads to the typical problem in Galois theory: finding the Galois group of a given polynomial. In this thesis, we investigate the inverse problem: for a specific Galois group, what irreducible polynomials occur. More specifically, we look at monic trinomials - polynomials with only three terms, having 1 as the leading coeficient. The first unresolved case of trinomials are of degree six and we will look specifically at trinomials of the form x⁶ + ax + b. We begin by investigating families of these trinomials that will result in Galois groups having a particular structure. From these families of trinomials, we can then make a final determination of individual Galois groups after eliminating any reducible possibilities. In the main calculations of this thesis, we investigate two parametric families of trinomials, one of which is given in [1]. From these families we completely characterize five of the possible sixteen Galois groups that can occur for sextic polynomials. In the notation of Butler and Mckay [2], these groups are 6T1, 6T2, 6T4, 6T5, and 6T6. In the final determination of these polynomials, rational points are found on genus 2 curves using a method known as elliptic Chabauty. We give an introduction to Galois theory followed by a brief explanation of the methods used to attain our results. We then discuss our results and proceed to prove them through the use of powerful software such as MAPLE (tm) and the Magma algebra system [3].