Abstract Algebra: A Comprehensive Introduction

Cover
Half-title
Series information
Title page
Copyright information
Dedication
Contents
Preface
1 A Refresher on the Integers
	1.1 Euclidean Division and the Greatest Common Divisor
	1.2 Primes and Unique Factorization
	1.3 Congruences
2 A First Look at Groups
	2.1 Basic Properties and Examples of Groups
		2.1.1 The Order of a Group and of Its Elements
	2.2 The Symmetric Group
		2.2.1 Equivalence Relations and Partitions
		2.2.2 The Orbits and Cycles of a Permutation
		2.2.3 Transpositions and the Parity of a Permutation
	2.3 Subgroups and Lagrange\'s Theorem
		2.3.1 Partitions of a Group by Right Cosets of a Subgroup
		2.3.2 Subgroups of Cyclic Groups
	2.4 Conjugation and Normal Subgroups
		2.4.1 Conjugates in S[sub(n)]
		2.4.2 Normal Subgroups of A[sub(n)]
		2.4.3 Normal Subgroups of the Quaternion Group
	2.5 Homomorphisms
		2.5.1 The Kernel and Image of a Homomorphism
		2.5.2 Isomorphisms
		2.5.3 Automorphisms of Cyclic Groups
	2.6 Quotient Groups
		2.6.1 An Important Theorem of Cauchy
		2.6.2 The First Isomorphism Theorem
		2.6.3 The Correspondence Theorem
	2.7 Products of Groups
		2.7.1 The External Product
		2.7.2 The Internal Product of Subgroups
	2.8 Finite Abelian Groups
		2.8.1 Groups of Prime Power Order
		2.8.2 The Primary Decomposition
		2.8.3 Putting the Bits Together
3 Groups Acting on Sets
	3.1 Definition and Some Illustrations of Actions
		3.1.1 A Group Action on Left Cosets
		3.1.2 Groups of Order 6
	3.2 Orbits and Stabilizers
		3.2.1 An Action for the Dihedral Groups
		3.2.2 The Symmetries of a Tetrahedron
	3.3 The Cauchy–Frobenius–Burnside Formula
		3.3.1 A Counting Technique Based on Burnside
	3.4 The Class Equation and Its Implications
		3.4.1 The Class Equation
		3.4.2 The Class Equation and p-Groups
	3.5 The Theorems of Cauchy and Sylow
		3.5.1 Cauchy\'s Theorem
		3.5.2 Sylow\'s Theorem
	3.6 Semi-direct Products
		3.6.1 Automorphism Actions of One Group on Another
		3.6.2 Internal and Semi-direct Products
		3.6.3 Groups of Order pq – the Full Description
	3.7 Solvable Groups
		3.7.1 The Second and Third Isomorphism Theorems
		3.7.2 Subgroups and Quotients of Solvable Groups
		3.7.3 Improving the Subnormal Chains for Solvable Groups
		3.7.4 Commutators and the Derived Series
	3.8 Breaking the Enigma
		3.8.1 The Design of the Enigma Machine
		3.8.2 The Mathematical Representation of the Enigma Machine
		3.8.3 How the Machine Was Used
		3.8.4 Finding the First Rotor Wiring
4 Basics on Rings – Mostly Commutative
	4.1 Terminology and Examples
		4.1.1 Compound Additions and Multiplications
		4.1.2 Subrings
	4.2 Units and Zero Divisors
		4.2.1 The Group of Units
		4.2.2 Zero Divisors
		4.2.3 Integral Domains and Fields
	4.3 Polynomials
		4.3.1 The Definition of Polynomial Rings
		4.3.2 Properties of the Degree
		4.3.3 Polynomials in Several Variables
		4.3.4 The Ring of Formal Power Series
	4.4 Homomorphisms and Ideals
		4.4.1 Ring Homomorphisms
		4.4.2 The Kernel
		4.4.3 Ideals
	4.5 Ideals in Z and in Polynomial Rings
		4.5.1 Finite Groups of Units in a Field
	4.6 Quotient Rings and the Isomorphism Theorem
		4.6.1 The First Isomorphism Theorem for Rings
		4.6.2 Computing the Euler Function
		4.6.3 The Correspondence Theorem
	4.7 Maximal and Prime Ideals
		4.7.1 Maximal Ideals and the Construction of Fields
		4.7.2 Existence of Maximal Ideals
		4.7.3 Prime Ideals and Integral Domains
		4.7.4 Building Fields from Polynomial Rings
	4.8 Fractions
		4.8.1 Localizations at Denominator Sets
		4.8.2 Uniqueness of Localizations
		4.8.3 Existence of Localizations
5 Primes and Unique Factorization
	5.1 Primes, Irreducibles and Factoring
	5.2 Principal Ideal and Noetherian Domains
		5.2.1 Noetherian Domains
	5.3 Euclidean Domains
	5.4 Gaussian Primes and Sums of Squares
	5.5 Greatest Common Divisors
	5.6 Polynomials over Unique Factorization Domains
		5.6.1 Gauss\' Lemma
		5.6.2 The Primes of A[X] and Its Unique Factorization
	5.7 Irreducible Polynomials
		5.7.1 The Rational Root Test
		5.7.2 Using a Natural Extension of Homomorphisms
		5.7.3 Eisenstein\'s Criterion
	5.8 Polynomials over Noetherian Rings
		5.8.1 Hilbert\'s Basis Theorem: a Source of Noetherian Rings
		5.8.2 Eisenstein\'s Criterion for Noetherian Domains
		5.8.3 Primes in Rings Coming from Algebraic Integers
		5.8.4 A Principal Ideal Domain that Is Not Euclidean
6 Algebraic Field Extensions
	6.1 Algebraic Elements and Degrees of Extensions
		6.1.1 The Degree of a Field Extension
		6.1.2 Algebraic Elements
		6.1.3 The Minimal Polynomial of an Algebraic Element
		6.1.4 A Basis for a Singly Generated Algebraic Extension
		6.1.5 Algebraic Elements with the Same Minimal Polynomial
		6.1.6 The Tower Theorem
		6.1.7 Fields Generated by Several Algebraic Elements
		6.1.8 The Algebraic Closure inside a Field Extension
		6.1.9 The Tower Theorem and Composite Fields
		6.1.10 The Tower Theorem Used in Conjunction with Eisenstein for Noetherian Domains
	6.2 Splitting Fields
		6.2.1 The Synthesis of Algebraic Elements
		6.2.2 What Is a Splitting Field?
		6.2.3 Lifting Isomorphisms to Splitting Fields
		6.2.4 The Uniqueness of Splitting Fields
		6.2.5 Finite Fields of Equal Size Are Isomorphic
		6.2.6 Splitting Fields Are Rich with Automorphisms
		6.2.7 Fixed Fields, F-Maps and F-Conjugates
		6.2.8 The Normality Theorem
		6.2.9 A Control on the Degree of a Splitting Field
		6.2.10 Quadratic Extensions Are Splitting Fields
	6.3 Separability
		6.3.1 Derivatives of Polynomials and Repeated Roots
		6.3.2 Separable Polynomials and the Derivative
		6.3.3 Finite Fields of All Possible Sizes
		6.3.4 Factoring X[sup(p[sup(n)])]-X in Z[sub(p)][X]
		6.3.5 Perfect Fields
	6.4 The Galois Group
		6.4.1 Roots, Generators and Galois Groups
		6.4.2 Some Examples of Galois Groups
	6.5 The Core of Galois Theory
		6.5.1 The Independence of Characters
		6.5.2 A Bound on the Order of a Galois Group
		6.5.3 The Fixed Field of an Automorphism Group
		6.5.4 Galois Extensions
		6.5.5 Artin\'s Theorem
		6.5.6 The Galois Correspondence
		6.5.7 The Fundamental Theorem of Galois Theory
		6.5.8 The Normality Connection
7 Applications of Galois Theory
	7.1 Three Classical Theorems
		7.1.1 The Primitive Generator Theorem
		7.1.2 The Fundamental Theorem of Algebra
		7.1.3 The Symmetric Function Theorem
	7.2 Special Extensions and Their Galois Groups
		7.2.1 The Galois Correspondence in Finite Fields
		7.2.2 The Galois Groups of Cubics
		7.2.3 Cyclotomic Extensions
	7.3 Solvability of Equations by Radicals
		7.3.1 Cardano\'s Formula
		7.3.2 Extensions by a Single Radical and Cyclic Galois Groups
		7.3.3 Radical Towers
		7.3.4 Solvable Polynomials Have Solvable Galois Groups
		7.3.5 Polynomials with Solvable Galois Groups Are Solvable
	7.4 Ruler and Compass Constructions
		7.4.1 Constructible Points, Lines and Circles
		7.4.2 Constructible Real Numbers
		7.4.3 Constructible Real Numbers and Field Towers
		7.4.4 Constructible Complex Numbers
	7.5 The Inverse Galois Problem over Q
8 Modules over Principal Ideal Domains
	8.1 The Language and Tools of Modules
		8.1.1 Examples of Modules
		8.1.2 Module Homomorphisms
		8.1.3 Submodules
		8.1.4 Free Modules
		8.1.5 Quotient Modules and the First Isomorphism Theorem
		8.1.6 Cyclic Modules and Annihilator Ideals
		8.1.7 Direct Sums of Submodules
		8.1.8 Free Modules Are Projective
	8.2 Modules over Integral Domains
		8.2.1 Torsion Elements and Modules
		8.2.2 Rank
	8.3 Modules over Principal Ideal Domains
		8.3.1 Splitting Torsion from Torsion-Free
		8.3.2 Torsion Modules over a Principal Ideal Domain
		8.3.3 Primary Modules over a Principal Ideal Domain
		8.3.4 Structure of Finitely Generated Modules over a PrincipalIdeal Domain
	8.4 Linear Algebra and Modules
9 Division Algorithms
	9.1 Well-Partial Orders
		9.1.1 Well-Ordered Sets: Total and Partial
		9.1.2 The Dickson Basis
		9.1.3 Extensions of Well-Partial Orders
	9.2 Gröbner Domains
		9.2.1 Gröbner Functions
		9.2.2 The Gröbner Basis of an Ideal
		9.2.3 Polynomials in Several Variables Are Gröbner Domains
		9.2.4 A Division Algorithm for Complete Reductions
	9.3 Buchberger\'s Algorithm
		9.3.1 Detecting Gröbner Bases via S-Polynomials
		9.3.2 Buchberger\'s Algorithm
	9.4 Applications of Gröbner Bases
		9.4.1 Gröbner Bases for Ideal Intersections
		9.4.2 Units and Zero Divisors in Finitely Generated Algebras
		9.4.3 Normal Forms
		9.4.4 Finite Varieties
		9.4.5 Hilbert\'s Nullstellensatz and Idempotents
Appendix A Infinite Sets
	A.1 Zorn\'s Lemma
		A.1.1 Choice Functions
		A.1.2 Chains of Subsets
		A.1.3 Chain Maximality in Partially Ordered Sets
		A.1.4 Choice Implies Zorn
		A.1.5 The Well-Ordering Principle
		A.1.6 Zorn Implies Choice
	A.2 The Size of Infinite Sets
		A.2.1 Infinite Sets and the Positive Integers
		A.2.2 Comparing Sets
		A.2.3 Countable Sets
		A.2.4 The Cantor–Schröder–Bernstein Theorem
		A.2.5 Total Ordering by Cardinality
		A.2.6 The Unboundedness of Cardinality
		A.2.7 A Bit of Cardinal Arithmetic
	A.3 The Algebraic Closure of a Field
Index