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Introduction to Abstract Algebra:
with Applications
Thomas W. Judson
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
Exercises
Sage
Sage Exercises
References and Suggested Readings
2
The Integers
Mathematical Induction
The Division Algorithm
Exercises
Programming Exercises
Sage
Sage Exercises
References and Suggested Readings
3
Modular Arithmetic
The Integers mod
\(n\)
Check digits
Chinese Remainder Theorem
Exercises
Programming Exercise
References and Suggested Readings
4
Rings
Rings
Integral Domains and Fields
Ring Homomorphisms
Fields of Fractions
Exercises
5
Polynomials
Polynomial Rings
The Division Algorithm
Irreducible Polynomials
Irreducibility Criteria
Exercises
Additional Exercises
Sage
Sage Exercises
6
Factorization in Integral Domains
Euclidean Domains
Unique Factorization Domains
Factorization of Gauss integers
Factorization in
\(D\lbrack x\rbrack\)
Exercises
Sage
References and Suggested Readings
7
Ideals
Ideals and Quotient Rings
Maximal and Prime Ideals
Principal Ideal Domains
Exercises
Sage
Sage Exercises
References and Suggested Readings
8
Groups
Symmetries
Definitions and Examples
Subgroups
Exercises
Sage
Sage Exercises
References and Suggested Readings
9
Cyclic Groups
Cyclic Subgroups
Multiplicative Group of Complex Numbers
The Method of Repeated Squares
Exercises
Programming Exercises
Sage
Sage Exercises
References and Suggested Readings
10
Permutation Groups
Definitions and Notation
Dihedral Groups
Exercises
Sage
Sage Exercises
11
Cosets and Lagrange's Theorem
Cosets
Lagrange's Theorem
Euler's Theorem
Exercises
Sage
Sage Exercises
12
Introduction to Cryptography
Private Key Cryptography
Public Key Cryptography
Exercises
Additional Exercises
Sage
Sage Exercises
References and Suggested Readings
13
Group Actions
Groups Acting on Sets
The Class Equation
Burnside's Counting Theorem
Exercises
Programming Exercise
Sage
Sage Exercises
References and Suggested Reading
Reference
A
GNU Free Documentation License
B
Hints to Selected Exercises
C
Notation
Index
Colophon
Authored in PreTeXt
Colophon
Colophon
This book was authored and produced with
PreTeXt
7
.
pretextbook.org