Preface Preface

In 1832, at the age of twenty, Évariste Galois was killed in a duel, having spent the night before writing down the mathematics he had developed over the previous few years. In that work he settled a question that had stood since antiquity: for which polynomial equations can a solution be written down using only the operations of arithmetic and the extraction of roots? Galois answered this by attaching to each polynomial a group, now called its Galois group, and showing that the equation is solvable by radicals exactly when this group has a particular structure. In doing so he did not just settle the question of the quintic; he introduced the idea that the symmetries of a mathematical object, organized as a group, could reveal its deepest properties. That idea, largely unrecognized in his lifetime, became one of the organizing principles of modern mathematics.

This text is an introduction to modern algebra built around that idea. We begin with the algebraic structures, groups, rings, and fields, that Galois’s insight requires, developing along the way the group actions and Sylow theory needed to understand the structure of groups, and the ring and field theory needed to understand polynomials and their roots. These threads come together in the final chapter, where we develop Galois theory itself and use it to answer the question that motivated Galois: exactly which polynomial equations are solvable by radicals. The goal throughout is to let this destination shape the journey, so that the algebra developed along the way is seen not as a list of definitions to memorize, but as the machinery required to prove one of the most striking theorems in mathematics.