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Abstract Algebra:
Theory and Applications
Thomas W. Judson
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Contents
Index
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Contents
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Front Matter
Colophon
Acknowledgements
Preface
1
Preliminaries
A Short Note on Proofs
Sets and Equivalence Relations
Reading Questions
Exercises
References and Suggested Readings
Sage
Sage Exercises
2
The Integers
Mathematical Induction
The Division Algorithm
Reading Questions
Exercises
Programming Exercises
References and Suggested Readings
Sage
Sage Exercises
3
Groups
Integer Equivalence Classes and Symmetries
Definitions and Examples
Subgroups
Reading Questions
Exercises
Additional Exercises: Detecting Errors
References and Suggested Readings
Sage
Sage Exercises
4
Cyclic Groups
Cyclic Subgroups
Multiplicative Group of Complex Numbers
The Method of Repeated Squares
Reading Questions
Exercises
Programming Exercises
References and Suggested Readings
Sage
Sage Exercises
5
Permutation Groups
Definitions and Notation
Dihedral Groups
Reading Questions
Exercises
Sage
Sage Exercises
6
Cosets and Lagrange's Theorem
Cosets
Lagrange's Theorem
Fermat's and Euler's Theorems
Reading Questions
Exercises
Sage
Sage Exercises
7
Introduction to Cryptography
Private Key Cryptography
Public Key Cryptography
Reading Questions
Exercises
Additional Exercises: Primality and Factoring
References and Suggested Readings
Sage
Sage Exercises
8
Algebraic Coding Theory
Error-Detecting and Correcting Codes
Linear Codes
Parity-Check and Generator Matrices
Efficient Decoding
Reading Questions
Exercises
Programming Exercises
References and Suggested Readings
Sage
Sage Exercises
9
Isomorphisms
Definition and Examples
Direct Products
Reading Questions
Exercises
Sage
Sage Exercises
10
Normal Subgroups and Factor Groups
Factor Groups and Normal Subgroups
The Simplicity of the Alternating Group
Reading Questions
Exercises
Sage
Sage Exercises
11
Homomorphisms
Group Homomorphisms
The Isomorphism Theorems
Reading Questions
Exercises
Additional Exercises: Automorphisms
Sage
Sage Exercises
12
Matrix Groups and Symmetry
Matrix Groups
Symmetry
Reading Questions
Exercises
References and Suggested Readings
Sage
Sage Exercises
13
The Structure of Groups
Finite Abelian Groups
Solvable Groups
Reading Questions
Exercises
Programming Exercises
References and Suggested Readings
Sage
Sage Exercises
14
Group Actions
Groups Acting on Sets
The Class Equation
Burnside's Counting Theorem
Reading Questions
Exercises
Programming Exercise
References and Suggested Reading
Sage
Sage Exercises
15
The Sylow Theorems
The Sylow Theorems
Examples and Applications
Reading Questions
Exercises
A Project
References and Suggested Readings
Sage
Sage Exercises
16
Rings
Rings
Integral Domains and Fields
Ring Homomorphisms and Ideals
Maximal and Prime Ideals
An Application to Software Design
Reading Questions
Exercises
Programming Exercise
References and Suggested Readings
Sage
Sage Exercises
17
Polynomials
Polynomial Rings
The Division Algorithm
Irreducible Polynomials
Reading Questions
Exercises
Additional Exercises: Solving the Cubic and Quartic Equations
Sage
Sage Exercises
18
Integral Domains
Fields of Fractions
Factorization in Integral Domains
Reading Questions
Exercises
References and Suggested Readings
Sage
Sage Exercises
19
Lattices and Boolean Algebras
Lattices
Boolean Algebras
The Algebra of Electrical Circuits
Reading Questions
Exercises
Programming Exercises
References and Suggested Readings
Sage
Sage Exercises
20
Vector Spaces
Definitions and Examples
Subspaces
Linear Independence
Reading Questions
Exercises
References and Suggested Readings
Sage
Sage Exercises
21
Fields
Extension Fields
Splitting Fields
Geometric Constructions
Reading Questions
Exercises
References and Suggested Readings
Sage
Sage Exercises
22
Finite Fields
Structure of a Finite Field
Polynomial Codes
Reading Questions
Exercises
Additional Exercises: Error Correction for
BCH
Codes
References and Suggested Readings
Sage
Sage Exercises
23
Galois Theory
Field Automorphisms
The Fundamental Theorem
Applications
Reading Questions
Exercises
References and Suggested Readings
Sage
Sage Exercises
Reference
A
GNU Free Documentation License
B
Hints and Answers to Selected Exercises
C
Notation
Index
Colophon
Authored in PreTeXt
Colophon
Colophon
This book was authored and produced with
PreTeXt
18
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pretextbook.org