Algebra: Polynomials, Galois Theory and Applications

Suitable for advanced undergraduates and graduate students in mathematics and computer science, this precise, self-contained treatment of Galois theory features detailed proofs and complete solutions to exercises. Originally published in French as Algèbre — Polynômes, théorie de Galois et applications informatiques, this 2017 Dover Aurora edition marks the volume’s first English-language publication.
The three-part treatment begins by providing the essential introduction to Galois theory. The second part is devoted to the algebraic, normal, and separable Galois extensions that constitute the center of the theory and examines abelian, cyclic, cyclotomic, and radical extensions. This section enables readers to acquire a comprehensive understanding of the Galois group of a polynomial. The third part deals with applications of Galois theory, including excellent discussions of several important real-world applications of these ideas, including cryptography and error-control coding theory. Symbolic computation via the Maple computer algebra system is incorporated throughout the text (though other software of symbolic computation could be used as well), along with a large number of very interesting exercises with full solutions.

Table of Contents

Part I Arithmetic, rings and polynomials
Chapter 1 Arithmetic and the symmetric group
Chapter 2 Rings and polynomials

Part II Galois theory
Chapter 3 Algebraic extensions
Chapter 4 Normal extensions and separable extensions
Chapter 5 Galois theory
Chapter 6 Abelian, cyclic, cyclotomic, radical extensions
Chapter 7 Galois group of a polynomial

Part III Applications
Chapter 8 Ruler and compass constructions
Chapter 9 Finite fields and applications
Chapter 10 Norm, trace and algebraic integers