Introduction to Galois Theory

This textbook provides an undergraduate introduction to Galois theory and its most notable applications.

Galois theory was born in the 19th century to study polynomial equations. Both powerful and elegant, this theory was at the origin of a substantial part of modern algebra and has since undergone considerable development. It remains an extremely active research subject and has found numerous applications beyond pure mathematics. In this book, the authors introduce Galois theory from a contemporary point of view. In particular, modern methods such as reduction modulo prime numbers and finite fields are introduced and put to use. Beyond the usual applications of ruler and compass constructions and solvability by radicals, the book also includes topics such as the transcendence of e and p, the inverse Galois problem, and infinite Galois theory.

Based on courses of the authors at the École Polytechnique, the book is aimed at students with a standard undergraduate background in (mostly linear) algebra. It includes a collection of exam questions in the form of review exercises, with detailed solutions.

1 Invitation to Galois Theory.- 2 Basic Concepts of Group Theory.- 3 Basic Concepts of Ring Theory.- 4 Basic Concepts of Algebras Over a Field.- 5 Finite Fields, Perfect Fields.- 6 The Galois Correspondence.- 7 Addendum:  Infinite Galois Correspondence.- 8 Cyclotomy and Constructibility.- 9 Solvability by Radicals.- 10 Reduction Modulo p.- 11 Complements.- 12 Review Exercises.- 13 Solutions to Exercises.

David Hernandez defended his PhD thesis in 2004 at the École Normale Supérieure and was thereafter a CNRS researcher. Since 2010 he is a full Professor at Université Paris Cité and has taught at the École Polytechnique. His research area is representation theory, currently in relation to cluster algebras, quiver varieties and quantum integral models. He was a member of the Institut Universitaire de France and a visiting scholar at UC Berkeley. He is the recipient of a Jacques Herbrand Prize, an ERC Consolidator grant and a France-Berkeley Fund Award. He is an invited speaker at the European Congress of Mathematicians in 2024.

Yves Laszlo defended his PhD thesis in 1989 at Université Paris-Sud. Thereafter he was a CNRS researcher, followed by full professorships at Sorbonne University, École Polytechnique and Paris-Saclay University. He has also been Vice President for Sciences at the École Normale Supérieure and Provost of both École Polytechnique and Institut Polytechnique de Paris. His research area is algebraic geometry.