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Abstract Algebra
Theory and Applications
Thomas W. Judson
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Front Matter
Colophon
Acknowledgements
Preface
1
Preliminaries
1.1
A Short Note on Proofs
1.1.1
Some Cautions and Suggestions
1.2
Sets and Equivalence Relations
1.2.1
Set Theory
1.2.2
Cartesian Products and Mappings
1.2.3
Equivalence Relations and Partitions
1.3
Reading Questions
1.4
Exercises
1.5
References and Suggested Readings
1.6
Sage
1.6.1
Executing Sage Commands
1.6.2
Immediate Help
1.6.3
Annotating Your Work
1.6.4
Lists
1.6.5
Lists of Integers
1.6.6
Saving and Sharing Your Work
1.7
Sage Exercises
2
The Integers
2.1
Mathematical Induction
2.2
The Division Algorithm
2.2.1
The Euclidean Algorithm
2.2.2
Prime Numbers
2.2.3
Historical Note
2.3
Reading Questions
2.4
Exercises
2.5
Programming Exercises
2.6
References and Suggested Readings
2.7
Sage
2.7.1
Division Algorithm
2.7.2
Greatest Common Divisor
2.7.3
Primes and Factoring
2.8
Sage Exercises
3
Groups
3.1
Integer Equivalence Classes and Symmetries
3.1.1
The Integers mod
\(n\)
3.1.2
Symmetries
3.2
Definitions and Examples
3.2.1
Basic Properties of Groups
3.2.2
Historical Note
3.3
Subgroups
3.3.1
Definitions and Examples
3.3.2
Some Subgroup Theorems
3.4
Reading Questions
3.5
Exercises
3.6
Additional Exercises: Detecting Errors
3.7
References and Suggested Readings
3.8
Sage
3.8.1
Integers mod n
3.8.2
Groups of symmetries
3.8.3
Quaternions
3.8.4
Subgroups
3.9
Sage Exercises
4
Cyclic Groups
4.1
Cyclic Subgroups
4.1.1
Subgroups of Cyclic Groups
4.2
Multiplicative Group of Complex Numbers
4.2.1
The Circle Group and the Roots of Unity
4.3
The Method of Repeated Squares
4.4
Reading Questions
4.5
Exercises
4.6
Programming Exercises
4.7
References and Suggested Readings
4.8
Sage
4.8.1
Infinite Cyclic Groups
4.8.2
Additive Cyclic Groups
4.8.3
Abstract Multiplicative Cyclic Groups
4.8.4
Cyclic Permutation Groups
4.8.5
Cayley Tables
4.8.6
Complex Roots of Unity
4.9
Sage Exercises
5
Permutation Groups
5.1
Definitions and Notation
5.1.1
Cycle Notation
5.1.2
Transpositions
5.1.3
The Alternating Groups
5.1.4
Historical Note
5.2
Dihedral Groups
5.2.1
The Motion Group of a Cube
5.3
Reading Questions
5.4
Exercises
5.5
Sage
5.5.1
Permutation Groups and Elements
5.5.2
Properties of Permutation Elements
5.5.3
Motion Group of a Cube
5.6
Sage Exercises
6
Cosets and Lagrange’s Theorem
6.1
Cosets
6.2
Lagrange’s Theorem
6.3
Fermat’s and Euler’s Theorems
6.3.1
Historical Note
6.4
Reading Questions
6.5
Exercises
6.6
Sage
6.6.1
Cosets
6.6.2
Subgroups
6.6.3
Subgroups of Cyclic Groups
6.6.4
Euler Phi Function
6.7
Sage Exercises
7
Introduction to Cryptography
7.1
Private Key Cryptography
7.2
Public Key Cryptography
7.2.1
The
RSA
Cryptosystem
7.2.2
Message Verification
7.2.3
Historical Note
7.3
Reading Questions
7.4
Exercises
7.5
Additional Exercises: Primality and Factoring
7.6
References and Suggested Readings
7.7
Sage
7.7.1
Constructing Keys
7.7.2
Signing and Encoding a Message
7.7.3
Decoding and Verifying a Message
7.8
Sage Exercises
8
Algebraic Coding Theory
8.1
Error-Detecting and Correcting Codes
8.1.1
Maximum-Likelihood Decoding
8.1.2
Block Codes
8.1.3
Historical Note
8.2
Linear Codes
8.2.1
Linear Codes
8.3
Parity-Check and Generator Matrices
8.4
Efficient Decoding
8.4.1
Coset Decoding
8.5
Reading Questions
8.6
Exercises
8.7
Programming Exercises
8.8
References and Suggested Readings
8.9
Sage
8.9.1
Constructing Linear Codes
8.9.2
Properties of Linear Codes
8.9.3
Decoding with a Linear Code
8.10
Sage Exercises
9
Isomorphisms
9.1
Definition and Examples
9.1.1
Cayley’s Theorem
9.1.2
Historical Note
9.2
Direct Products
9.2.1
External Direct Products
9.2.2
Internal Direct Products
9.3
Reading Questions
9.4
Exercises
9.5
Sage
9.5.1
Isomorphism Testing
9.5.2
Classifying Finite Groups
9.5.3
Internal Direct Products
9.6
Sage Exercises
10
Normal Subgroups and Factor Groups
10.1
Factor Groups and Normal Subgroups
10.1.1
Normal Subgroups
10.1.2
Factor Groups
10.2
The Simplicity of the Alternating Group
10.2.1
Historical Note
10.3
Reading Questions
10.4
Exercises
10.5
Sage
10.5.1
Multiplying Cosets
10.5.2
Sage Methods for Normal Subgroups
10.6
Sage Exercises
11
Homomorphisms
11.1
Group Homomorphisms
11.2
The Isomorphism Theorems
11.3
Reading Questions
11.4
Exercises
11.5
Additional Exercises: Automorphisms
11.6
Sage
11.6.1
Homomorphisms
11.7
Sage Exercises
12
Matrix Groups and Symmetry
12.1
Matrix Groups
12.1.1
Some Facts from Linear Algebra
12.1.2
The General and Special Linear Groups
12.1.3
The Orthogonal Group
\(O(n)\)
12.2
Symmetry
12.2.1
The Wallpaper Groups
12.2.2
Historical Note
12.3
Reading Questions
12.4
Exercises
12.5
References and Suggested Readings
12.6
Sage
12.7
Sage Exercises
13
The Structure of Groups
13.1
Finite Abelian Groups
13.2
Solvable Groups
13.3
Reading Questions
13.4
Exercises
13.5
Programming Exercises
13.6
References and Suggested Readings
13.7
Sage
13.7.1
Classification of Finite Groups
13.7.2
Groups of Small Order as Permutation Groups
13.8
Sage Exercises
14
Group Actions
14.1
Groups Acting on Sets
14.2
The Class Equation
14.3
Burnside’s Counting Theorem
14.3.1
A Geometric Example
14.3.2
Switching Functions
14.3.3
Historical Note
14.4
Reading Questions
14.5
Exercises
14.6
Programming Exercise
14.7
References and Suggested Reading
14.8
Sage
14.8.1
Conjugation as a Group Action
14.8.2
Graph Automorphisms
14.9
Sage Exercises
15
The Sylow Theorems
15.1
The Sylow Theorems
15.1.1
Historical Note
15.2
Examples and Applications
15.2.1
Finite Simple Groups
15.3
Reading Questions
15.4
Exercises
15.5
A Project
15.6
References and Suggested Readings
15.7
Sage
15.7.1
Sylow Subgroups
15.7.2
Normalizers
15.7.3
Finite Simple Groups
15.7.4
GAP
Console and Interface
15.8
Sage Exercises
16
Rings
16.1
Rings
16.2
Integral Domains and Fields
16.3
Ring Homomorphisms and Ideals
16.4
Maximal and Prime Ideals
16.4.1
Historical Note
16.5
An Application to Software Design
16.6
Reading Questions
16.7
Exercises
16.8
Programming Exercise
16.9
References and Suggested Readings
16.10
Sage
16.10.1
Creating Rings
16.10.2
Properties of Rings
16.10.3
Quotient Structure
16.10.4
Ring Homomorphisms
16.11
Sage Exercises
17
Polynomials
17.1
Polynomial Rings
17.2
The Division Algorithm
17.3
Irreducible Polynomials
17.3.1
Ideals in
\(F\lbrack x \rbrack\)
17.3.2
Historical Note
17.4
Reading Questions
17.5
Exercises
17.6
Additional Exercises: Solving the Cubic and Quartic Equations
17.7
Sage
17.7.1
Polynomial Rings and their Elements
17.7.2
Irreducible Polynomials
17.7.3
Polynomials over Fields
17.8
Sage Exercises
18
Integral Domains
18.1
Fields of Fractions
18.2
Factorization in Integral Domains
18.2.1
Principal Ideal Domains
18.2.2
Euclidean Domains
18.2.3
Factorization in
\(D\lbrack x \rbrack\)
18.2.4
Historical Note
18.3
Reading Questions
18.4
Exercises
18.5
References and Suggested Readings
18.6
Sage
18.6.1
Field of Fractions
18.6.2
Prime Subfields
18.6.3
Integral Domains
18.6.4
Principal Ideals
18.7
Sage Exercises
19
Lattices and Boolean Algebras
19.1
Lattices
19.1.1
Partially Ordered Sets
19.2
Boolean Algebras
19.2.1
Finite Boolean Algebras
19.3
The Algebra of Electrical Circuits
19.3.1
Historical Note
19.4
Reading Questions
19.5
Exercises
19.6
Programming Exercises
19.7
References and Suggested Readings
19.8
Sage
19.8.1
Creating Partially Ordered Sets
19.8.2
Properties of a Poset
19.8.3
Lattices
19.9
Sage Exercises
20
Vector Spaces
20.1
Definitions and Examples
20.2
Subspaces
20.3
Linear Independence
20.4
Reading Questions
20.5
Exercises
20.6
References and Suggested Readings
20.7
Sage
20.7.1
Vector Spaces
20.7.2
Subspaces
20.7.3
Linear Independence
20.7.4
Abstract Vector Spaces
20.7.5
Linear Algebra
20.8
Sage Exercises
21
Fields
21.1
Extension Fields
21.1.1
Algebraic Elements
21.1.2
Algebraic Closure
21.2
Splitting Fields
21.3
Geometric Constructions
21.3.1
Constructible Numbers
21.3.2
Doubling the Cube and Squaring the Circle
21.3.3
Trisecting an Angle
21.3.4
Historical Note
21.4
Reading Questions
21.5
Exercises
21.6
References and Suggested Readings
21.7
Sage
21.7.1
Number Fields
21.7.2
Relative and Absolute Number Fields
21.7.3
Splitting Fields
21.7.4
Algebraic Numbers
21.7.5
Geometric Constructions
21.8
Sage Exercises
22
Finite Fields
22.1
Structure of a Finite Field
22.2
Polynomial Codes
22.2.1
Polynomial Codes
22.2.2
BCH
Codes
22.3
Reading Questions
22.4
Exercises
22.5
Additional Exercises: Error Correction for
BCH
Codes
22.6
References and Suggested Readings
22.7
Sage
22.7.1
Creating Finite Fields
22.7.2
Logarithms in Finite Fields
22.8
Sage Exercises
23
Galois Theory
23.1
Field Automorphisms
23.1.1
Separable Extensions
23.2
The Fundamental Theorem
23.2.1
Historical Note
23.3
Applications
23.3.1
Solvability by Radicals
23.3.2
Insolvability of the Quintic
23.3.3
The Fundamental Theorem of Algebra
23.4
Reading Questions
23.5
Exercises
23.6
References and Suggested Readings
23.7
Sage
23.7.1
Galois Groups
23.7.2
Fixed Fields
23.7.3
Galois Correspondence
23.7.4
Normal Extensions
23.8
Sage Exercises
Reference
A
GNU Free Documentation License
B
Hints and Answers to Selected Exercises
C
Notation
Index
Colophon
Section
12.6
Sage
There is no Sage material for this chapter.
