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Abstract Algebra:
Theory and Applications, Remixed!
Thomas W. Judson
Contents
Index
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Contents
Prev
Up
Next
Front Matter
Colophon
Acknowledgements
Preface to the Remixed Version
Original Preface
I
Groups
1
Preliminaries
A Short Note on Proofs
Sets and Functions
Equivalence Relations and Partitions
Mathematical Induction
Additional Exercises
References and Suggested Readings
2
Groups
Integer Equivalence Classes and Symmetries
Definitions and Examples
Basic Properties of Groups
Subgroups
Cyclic Subgroups
Permutation Groups
Additional Exercises
Additional Exercises: Detecting Errors
References and Suggested Readings
3
Cosets and Factor Groups
Cosets
Factor Groups and Normal Subgroups
Exercises
4
Isomorphisms and Homomorphisms
Definition and Examples
Group Homomorphisms
The Isomorphism Theorems
Additional Exercises
Additional Exercises: Automorphisms
II
Rings and Fields
5
Rings
Rings
Integral Domains and Fields
Ring Homomorphisms and Ideals
Maximal and Prime Ideals
An Application to Software Design
Additional Exercises
Programming Exercise
References and Suggested Readings
6
Polynomials
The Integers
Polynomial Rings
The Division Algorithm
Irreducible Polynomials
Factoring over \(\C\) and \(\R\)
Additional Exercises
Additional Exercises: The Euclidean Algorithm
Additional Exercises: Solving the Cubic and Quartic Equations
7
Integral Domains
Fields of Fractions
Factorization in Integral Domains
Reading Questions
Exercises
References and Suggested Readings
8
Fields
Geometric Constructions
A Review of Rings and Fields
Extension Fields
Algebraic Elements
A Review of Vector Spaces
Extension Fields as Vector Spaces
What's Not Constructible
9
Galois Theory
Splitting Fields
Field Automorphisms
The Fundamental Theorem
III
More Group Theory
10
Some of the Structure of Groups
Isomorphisms and Cayley's Theorem
Permutation Groups
Solvable Groups
The group \(S_5\)
Applications: Galois Groups
11
More of the Structure of Groups
Cyclic Groups and Orders of Elements
Lagrange's Theorem and its consequences
Direct Products
Finite Abelian Groups
12
Introduction to Cryptography
Private Key Cryptography
Public Key Cryptography
Activity: RSA Cryptography Lab
The Method of Repeated Squares
Exercises
Additional Exercises: Primality and Factoring
References and Suggested Readings
13
Group Actions
Groups Acting on Sets
The Class Equation
Burnside's Counting Theorem
Exercises
Programming Exercise
References and Suggested Reading
14
The Sylow Theorems
The Sylow Theorems
Examples and Applications
Reading Questions
Exercises
A Project
References and Suggested Readings
IV
Additional Topics
15
Matrix Groups and Symmetry
Matrix Groups
Symmetry
Reading Questions
Exercises
References and Suggested Readings
16
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
17
Finite Fields
Structure of a Finite Field
Polynomial Codes
Reading Questions
Exercises
Additional Exercises: Error Correction for
BCH
Codes
References and Suggested Readings
18
Lattices and Boolean Algebras
Lattices
Boolean Algebras
The Algebra of Electrical Circuits
Reading Questions
Exercises
Programming Exercises
References and Suggested Readings
Reference
A
GNU Free Documentation License
B
Hints and Answers to Selected Exercises
C
Notation
Index
Colophon
Authored in PreTeXt
Abstract Algebra:
Theory and Applications, Remixed!
Thomas W. Judson
Department of Mathematics and Statistics
Stephen F. Austin State University
[email protected]
Remixed Version
Oscar Levin
School of Mathematical Sciences
University of Northern Colorado
[email protected]
Sage Exercises for Abstract Algebra
Robert A. Beezer
Department of Mathematics and Computer Science
University of Puget Sound
[email protected]
Traducción al español
Antonio Behn
Departamento de Matemáticas, Facultad de Ciencias
Universidad de Chile
[email protected]
August 19, 2021
Colophon
Acknowledgements
Preface to the Remixed Version
Original Preface
