This is a course in modern abstract algebra, focusing on Galois theory. This is a beautiful subject, which uses group theory to study symmetries of solutions to polynomial equations.

You are all familiar with the famous quadratic formula. There also exist cubic and quartic formulas, though they are significantly more complicated. For a general quintic polynomial, however, it turns out that there does not exist any analogous formula for its roots. By this I do not simply mean that nobody has been able to find such a formula. I mean that, in a precise and provable sense, no such formula can ever exist! This was first proved by Galois, a French revolutionary who spent time in and out of prison before dying in a duel at the age of 20.

In this course we will develop the ideas and techniques necessary to prove the insolubility of the quintic. Along the way, we will encounter many of the cornerstones of abstract algebra: rings, fields, polynomials, and Galois groups.