Fearless Symmetry: Exposing the hidden patterns of numbers by Avner Ash and Robert Gross
An awesome maths book that covers the material needed for a proof of Fermat’s Last Theorem. The book is written for the amateur mathematician, with lots of examples demonstrating the various definitions and omitting most of the proofs. In part one, the book starts with group theory, and then moves onto equations and varieties and on to quadratic reciprocity. Part two covers Galois theory, the field of algebraic numbers, matrices and group representations. It then moves on to cover the Galois group of a polynomial, conjugacy classes and Frobenius elements. Part three takes this material and moves on to the weak and strong reciprocity laws and sketches how the material can be put together to get a proof of Fermat’s Last Theorem.
The book tries to make all of the material very accessible. Definitions and their motivation are explained very clearly with simple, well-explained examples. A very well written book.