Abstract Algebra: With Applications to Galois Theory, Algebraic Geometry, Representation Theory and Cryptography

✍ Scribed by Gerhard Rosenberger , Annika Schürenberg , Leonard Wienke

• Publisher: De Gruyter
• Year: 2024
• Tongue: English
• Leaves: 423
• Series: De Gruyter Textbook
• Edition: 3
• Category: Library

✦ Synopsis

Abstract algebra is the study of algebraic structures like groups, rings and fields. This book provides an account of the theoretical foundations including applications to Galois Theory, Algebraic Geometry and Representation Theory. It implements the pedagogic approach to conveying algebra from the perspective of rings. The 3rd edition provides a revised and extended versions of the chapters on Algebraic Cryptography and Geometric Group Theory.

• Uses a modern approach to introduce algebra via rings and integers rather than via group theory.

• Covers unique topics such as Algebraic Geometry and cryptography.

• Includes important recent applications and accompanying exercises.

✦ Table of Contents

cover
Preface
Contents
1 Groups, Rings and Fields
2 Maximal and Prime Ideals
3 Prime Elements and Unique Factorization Domains
4 Polynomials and Polynomial Rings
5 Field Extensions
6 Field Extensions and Compass and Straightedge Constructions
7 Kronecker’s Theorem and Algebraic Closures
8 Splitting Fields and Normal Extensions
9 Groups, Subgroups and Examples
10 Normal Subgroups, Factor Groups and Direct Products
11 Symmetric and Alternating Groups
12 Solvable Groups
13 Group Actions and the Sylow Theorems
14 Free Groups and Group Presentations
15 Finite Galois Extensions
16 Separable Field Extensions
17 Applications of Galois Theory
18 The Theory of Modules
19 Finitely Generated Abelian Groups
20 Integral and Transcendental Extensions
21 The Hilbert Basis Theorem and the Nullstellensatz
22 Algebras and Group Representations
23 Algebraic Cryptography
24 Non-Commutative Group Based Cryptography
Bibliography
Index

✦ Subjects

Groups, Rings, Fields, Galois Theory, Modules, Nullstellensatz, Algebraic Cryptography