Skip to main content
Abstract Algebra:
Theory and Applications
Thomas W. Judson
Contents
Index
Search Book
close
Search Results:
No results.
Prev
Up
Next
\(\newcommand{\identity}{\mathrm{id}} \newcommand{\notdivide}{\nmid} \newcommand{\notsubset}{\not\subset} \newcommand{\lcm}{\operatorname{lcm}} \newcommand{\gf}{\operatorname{GF}} \newcommand{\inn}{\operatorname{Inn}} \newcommand{\aut}{\operatorname{Aut}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\cis}{\operatorname{cis}} \newcommand{\chr}{\operatorname{char}} \newcommand{\Null}{\operatorname{Null}} \newcommand{\transpose}{\text{t}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \definecolor{fillinmathshade}{gray}{0.9} \newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}} \)
Front Matter
Colophon
Acknowledgements
Preface
1
Preliminaries
1.1
A Short Note on Proofs
Some Cautions and Suggestions
1.2
Sets and Equivalence Relations
Set Theory
Cartesian Products and Mappings
Equivalence Relations and Partitions
1.3
Reading Questions
1.4
Exercises
1.5
References and Suggested Readings
1.6
Sage
Executing Sage Commands
Immediate Help
Annotating Your Work
Lists
Lists of Integers
Saving and Sharing Your Work
1.7
Sage Exercises
2
The Integers
2.1
Mathematical Induction
2.2
The Division Algorithm
The Euclidean Algorithm
Prime Numbers
Historical Note
2.3
Reading Questions
2.4
Exercises
2.5
Programming Exercises
2.6
References and Suggested Readings
2.7
Sage
Division Algorithm
Greatest Common Divisor
Primes and Factoring
2.8
Sage Exercises
3
Groups
3.1
Integer Equivalence Classes and Symmetries
The Integers mod
\(n\)
Symmetries
3.2
Definitions and Examples
Basic Properties of Groups
Historical Note
3.3
Subgroups
Definitions and Examples
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
Integers mod n
Groups of symmetries
Quaternions
Subgroups
3.9
Sage Exercises
4
Cyclic Groups
4.1
Cyclic Subgroups
Subgroups of Cyclic Groups
4.2
Multiplicative Group of Complex Numbers
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
Infinite Cyclic Groups
Additive Cyclic Groups
Abstract Multiplicative Cyclic Groups
Cyclic Permutation Groups
Cayley Tables
Complex Roots of Unity
4.9
Sage Exercises
5
Permutation Groups
5.1
Definitions and Notation
Cycle Notation
Transpositions
The Alternating Groups
Historical Note
5.2
Dihedral Groups
The Motion Group of a Cube
5.3
Reading Questions
5.4
Exercises
5.5
Sage
Permutation Groups and Elements
Properties of Permutation Elements
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
Historical Note
6.4
Reading Questions
6.5
Exercises
6.6
Sage
Cosets
Subgroups
Subgroups of Cyclic Groups
Euler Phi Function
6.7
Sage Exercises
7
Introduction to Cryptography
7.1
Private Key Cryptography
7.2
Public Key Cryptography
The
RSA
Cryptosystem
Message Verification
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
Constructing Keys
Signing and Encoding a Message
Decoding and Verifying a Message
7.8
Sage Exercises
8
Algebraic Coding Theory
8.1
Error-Detecting and Correcting Codes
Maximum-Likelihood Decoding
Block Codes
Historical Note
8.2
Linear Codes
Linear Codes
8.3
Parity-Check and Generator Matrices
8.4
Efficient Decoding
Coset Decoding
8.5
Reading Questions
8.6
Exercises
8.7
Programming Exercises
8.8
References and Suggested Readings
8.9
Sage
Constructing Linear Codes
Properties of Linear Codes
Decoding with a Linear Code
8.10
Sage Exercises
9
Isomorphisms
9.1
Definition and Examples
Cayley’s Theorem
Historical Note
9.2
Direct Products
External Direct Products
Internal Direct Products
9.3
Reading Questions
9.4
Exercises
9.5
Sage
Isomorphism Testing
Classifying Finite Groups
Internal Direct Products
9.6
Sage Exercises
10
Normal Subgroups and Factor Groups
10.1
Factor Groups and Normal Subgroups
Normal Subgroups
Factor Groups
10.2
The Simplicity of the Alternating Group
Historical Note
10.3
Reading Questions
10.4
Exercises
10.5
Sage
Multiplying Cosets
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
Homomorphisms
11.7
Sage Exercises
12
Matrix Groups and Symmetry
12.1
Matrix Groups
Some Facts from Linear Algebra
The General and Special Linear Groups
The Orthogonal Group
\(O(n)\)
12.2
Symmetry
The Wallpaper Groups
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
Classification of Finite Groups
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
A Geometric Example
Switching Functions
Historical Note
14.4
Reading Questions
14.5
Exercises
14.6
Programming Exercise
14.7
References and Suggested Reading
14.8
Sage
Conjugation as a Group Action
Graph Automorphisms
14.9
Sage Exercises
15
The Sylow Theorems
15.1
The Sylow Theorems
Historical Note
15.2
Examples and Applications
Finite Simple Groups
15.3
Reading Questions
15.4
Exercises
15.5
A Project
15.6
References and Suggested Readings
15.7
Sage
Sylow Subgroups
Normalizers
Finite Simple Groups
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
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
Creating Rings
Properties of Rings
Quotient Structure
Ring Homomorphisms
16.11
Sage Exercises
17
Polynomials
17.1
Polynomial Rings
17.2
The Division Algorithm
17.3
Irreducible Polynomials
Ideals in
\(F\lbrack x \rbrack\)
Historical Note
17.4
Reading Questions
17.5
Exercises
17.6
Additional Exercises: Solving the Cubic and Quartic Equations
17.7
Sage
Polynomial Rings and their Elements
Irreducible Polynomials
Polynomials over Fields
17.8
Sage Exercises
18
Integral Domains
18.1
Fields of Fractions
18.2
Factorization in Integral Domains
Principal Ideal Domains
Euclidean Domains
Factorization in
\(D\lbrack x \rbrack\)
Historical Note
18.3
Reading Questions
18.4
Exercises
18.5
References and Suggested Readings
18.6
Sage
Field of Fractions
Prime Subfields
Integral Domains
Principal Ideals
18.7
Sage Exercises
19
Lattices and Boolean Algebras
19.1
Lattices
Partially Ordered Sets
19.2
Boolean Algebras
Finite Boolean Algebras
19.3
The Algebra of Electrical Circuits
Historical Note
19.4
Reading Questions
19.5
Exercises
19.6
Programming Exercises
19.7
References and Suggested Readings
19.8
Sage
Creating Partially Ordered Sets
Properties of a Poset
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
Vector Spaces
Subspaces
Linear Independence
Abstract Vector Spaces
Linear Algebra
20.8
Sage Exercises
21
Fields
21.1
Extension Fields
Algebraic Elements
Algebraic Closure
21.2
Splitting Fields
21.3
Geometric Constructions
Constructible Numbers
Doubling the Cube and Squaring the Circle
Trisecting an Angle
Historical Note
21.4
Reading Questions
21.5
Exercises
21.6
References and Suggested Readings
21.7
Sage
Number Fields
Relative and Absolute Number Fields
Splitting Fields
Algebraic Numbers
Geometric Constructions
21.8
Sage Exercises
22
Finite Fields
22.1
Structure of a Finite Field
22.2
Polynomial Codes
Polynomial Codes
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
Creating Finite Fields
Logarithms in Finite Fields
22.8
Sage Exercises
23
Galois Theory
23.1
Field Automorphisms
Separable Extensions
23.2
The Fundamental Theorem
Historical Note
23.3
Applications
Solvability by Radicals
Insolvability of the Quintic
The Fundamental Theorem of Algebra
23.4
Reading Questions
23.5
Exercises
23.6
References and Suggested Readings
23.7
Sage
Galois Groups
Fixed Fields
Galois Correspondence
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.