This carefully written textbook offers a thorough introduction to abstract algebra, covering the fundamentals of groups, rings and fields.
The first two chapters present preliminary topics such as properties of the integers and equivalence relations. The author then explores the first major algebraic structure, the group, progressing as far as the Sylow theorems and the classification of finite abelian groups. An introduction to ring theory follows, leading to a discussion of fields and polynomials that includes sections on splitting fields and the construction of finite fields. The final part contains applications to public key cryptography as well as classical straightedge and compass constructions.
Explaining key topics at a gentle pace, this book is aimed at undergraduate students. It assumes no prior knowledge of the subject and contains over 500 exercises, half of which have detailed solutions provided.
Author(s): Gregory T Lee
Series: Springer Undergraduate Mathematics Series
Publisher: Springer
Year: 2018
Language: English
Pages: 293
Tags: modern algebra
Preface
Contents
Part I Preliminaries
1 Relations and Functions
1.1 Sets and Set Operations
1.2 Relations
1.3 Equivalence Relations
1.4 Functions
2 The Integers and Modular Arithmetic
2.1 Induction and Well Ordering
2.2 Divisibility
2.3 Prime Factorization
2.4 Properties of the Integers
2.5 Modular Arithmetic
Part II Groups
3 Introduction to Groups
3.1 An Important Example
3.2 Groups
3.3 A Few Basic Properties
3.4 Powers and Orders
3.5 Subgroups
3.6 Cyclic Groups
3.7 Cosets and Lagrange's Theorem
4 Factor Groups and Homomorphisms
4.1 Normal Subgroups
4.2 Factor Groups
4.3 Homomorphisms
4.4 Isomorphisms
4.5 The Isomorphism Theorems for Groups
4.6 Automorphisms
5 Direct Products and the Classification of Finite Abelian Groups
5.1 Direct Products
5.2 The Fundamental Theorem of Finite Abelian Groups
5.3 Elementary Divisors and Invariant Factors
5.4 A Word About Infinite Abelian Groups
6 Symmetric and Alternating Groups
6.1 The Symmetric Group and Cycle Notation
6.2 Transpositions and the Alternating Group
6.3 The Simplicity of the Alternating Group
7 The Sylow Theorems
7.1 Normalizers and Centralizers
7.2 Conjugacy and the Class Equation
7.3 The Three Sylow Theorems
7.4 Applying the Sylow Theorems
7.5 Classification of the Groups of Small Order
Part III Rings
8 Introduction to Rings
8.1 Rings
8.2 Basic Properties of Rings
8.3 Subrings
8.4 Integral Domains and Fields
8.5 The Characteristic of a Ring
9 Ideals, Factor Rings and Homomorphisms
9.1 Ideals
9.2 Factor Rings
9.3 Ring Homomorphisms
9.4 Isomorphisms and Automorphisms
9.5 Isomorphism Theorems for Rings
9.6 Prime and Maximal Ideals
10 Special Types of Domains
10.1 Polynomial Rings
10.2 Euclidean Domains
10.3 Principal Ideal Domains
10.4 Unique Factorization Domains
Reference
Part IV Fields and Polynomials
11 Irreducible Polynomials
11.1 Irreducibility and Roots
11.2 Irreducibility over the Rationals
11.3 Irreducibility over the Real and Complex Numbers
11.4 Irreducibility over Finite Fields
Reference
12 Vector Spaces and Field Extensions
12.1 Vector Spaces
12.2 Basis and Dimension
12.3 Field Extensions
12.4 Splitting Fields
12.5 Applications to Finite Fields
Reference
Part V Applications
13 Public Key Cryptography
13.1 Private Key Cryptography
13.2 The RSA Scheme
14 Straightedge and Compass Constructions
14.1 Three Ancient Problems
14.2 The Connection to Field Extensions
14.3 Proof of the Impossibility of the Problems
A The Complex Numbers
Appendix B Matrix Algebra
Appendix Solutions
Index
