Abstract Algebra via Numbers

Preface
Set-Theoretic Preliminaries
Contents
About the Author
1 Number Theory
	1.1 Decomposition of Natural Numbers
	1.2 Mathematical Induction
	1.3 The Well-Ordering Principle
	1.4 Relative Primeness and Greatest Common Divisors
	1.5 The Euclidean Algorithm
	1.6 Newton\'s Binomial Formula
	1.7 Infinity of Primes
	1.8 Primes in Arithmetic Progression
	1.9 The Function π(x)
	1.10 Congruence
	1.11 Arithmetic Functions
	1.12 Primitive Roots
	1.13 Quadratic Reciprocity
	1.14 Certain Classes of Numbers
	1.15 Diophantine Equations
	1.16 Sums of Squares
	1.17 Fibonacci Numbers
2 Construction of Numbers
	2.1 Peano\'s Axioms
	2.2 The Integers Constructed from the Natural Numbers
	2.3 From the Integers to the Rational Numbers
	2.4 Finite Simple Continued Fractions
	2.5 Construction of the Real Numbers
	2.6 The Least Upper Bound Property
	2.7 Decimal Expansions
	2.8 Infinite Continued Fractions
	2.9 Pell\'s Equation
	2.10 Complex Numbers
	2.11 Absolute Values and p-Adic Numbers
	2.12 Cardinality
3 Linear Algebra
	3.1 Vector Spaces
	3.2 Linear Basis
	3.3 Linear Transformations
	3.4 Matrices
	3.5 Systems of Linear Equations
	3.6 Permutations
	3.7 Determinants
	3.8 Eigenvalues and Eigenvectors
	3.9 Jordan Canonical Form
	3.10 Dual Spaces, Inner Products and Tensor Products
4 Groups
	4.1 Groups and Semigroups
	4.2 Subgroups
	4.3 Generators
	4.4 Cosets and Lagrange\'s Theorem
	4.5 Morphisms
	4.6 Normal Subgroups
	4.7 Cyclic Groups
	4.8 Normalizers and Centralizers
	4.9 Correspondences
	4.10 More Isomorphism Theorems
	4.11 Permutation Groups
	4.12 Symmetries
	4.13 Automorphisms
	4.14 Semidirect Products
	4.15 The General and Special Linear Group
	4.16 Inner Products and Linear Subgroups
	4.17 Actions
	4.18 Spheres, Projective Spaces and Grassmannians
	4.19 Groups of Prime Power Orders
	4.20 Cauchy\'s Theorems and Sylow\'s First Theorem
	4.21 Sylow\'s Second and Third Theorems
	4.22 Some Examples
	4.23 Groups with Order the Product of Two Primes
	4.24 Normal Series
	4.25 The Theorem of Schreier
	4.26 Solvable Groups
	4.27 Nilpotent Groups
	4.28 Simplicity of the Alternating Group
	4.29 Transfer Homomorphisms
	4.30 Finitely Generated Abelian Groups
	4.31 Free Groups
	4.32 An Example of a Free Product
	4.33 Generators and Relations
	4.34 Ordered Groups
	4.35 Groups in Algebraic Topology
5 Representations of Finite Groups
	5.1 Basic Definitions
	5.2 Regular Functions
	5.3 New Representations from Old Ones
	5.4 Decomposition Into Irreducibles
	5.5 Haar Integral
	5.6 Regular Representation
	5.7 Schur\'s Lemma
	5.8 Characters of Abelian Groups
	5.9 Fourier Analysis
	5.10 Orthogonality Relations
	5.11 Three Auxiliary Representations
	5.12 Characters of Representations
	5.13 Group Algebra
	5.14 Quadratic Reciprocity from Fourier Analysis
	5.15 The Character Table for S3
	5.16 Induced Representations
	5.17 Reciprocity
	5.18 Mackey Theory
	5.19 Characters of Induced Representations
6 Rings
	6.1 Basic Definitions
	6.2 Prime Subfields and Characteristics
	6.3 Examples of Non-commutative Rings
	6.4 Group Rings
	6.5 Polynomial Rings
	6.6 Laurent Series and Power Series
	6.7 Ideals
	6.8 Quotient Rings and Homomorphisms
	6.9 Rings with Generators and Relations
	6.10 Twisted Group Rings
	6.11 Simple Rings and Maximal Ideals
	6.12 Euclidean Domains and Principal Ideal Domains
	6.13 Prime Ideals and Irreducible Ideals
	6.14 Unique Factorization in a PID
	6.15 Unique Factorization Domains
7 Field Extensions
	7.1 Roots and Reducible Polynomials
	7.2 Algebraic Extensions
	7.3 Algebraic Closures
	7.4 Ruler and Compass
	7.5 Splitting Fields and Normal Extensions
	7.6 Multiple Roots
	7.7 Finite Fields
	7.8 Separable Extensions
8 Galois Theory
	8.1 Automorphisms and Fixed Fields
	8.2 The Galois Group of a Polynomial
	8.3 The Fundamental Theorem in Galois Theory
	8.4 Proof of the Fundamental Theorem of Algebra
	8.5 Primitive Roots and Cyclotomic Polynomials
	8.6 Constructable Polygons
	8.7 Cyclic Extensions
	8.8 Polynomials Solvable by Radicals
	8.9 Symmetric Functions
	8.10 Cubic and Quartic Equations
9 Modules
	9.1 Basics
	9.2 Exactness
	9.3 Projectivity
	9.4 Injectivity
	9.5 Tensor Products and Bimodules
	9.6 Diagram Chase
	9.7 Flatness
	9.8 Duals
	9.9 Modules over PID\'s
	9.10 Torsion Modules over PID\'s
	9.11 Smith Normal Form
	9.12 Applications to Linear Algebra
	9.13 Generalized Jordan Blocks
	9.14 The Jordan–Chevalley Decomposition
	9.15 Semisimple Modules
	9.16 Density
	9.17 Semisimple Rings
	9.18 Noetherian and Artinian Modules
	9.19 Nilpotence
	9.20 The Jacobson Radical
	9.21 The Wedderburn Radical
	9.22 Radicals Under Change of Rings
	9.23 Radicals of Polynomial Rings
	9.24 Radicals of Groups Rings
	9.25 Units in Group Rings
	9.26 Division Rings
10 Appendix
	10.1 The Function π(x) for Large x
	10.2 The Riemann Zeta Function
	10.3 Bernoulli Numbers
	10.4 Transcendentality of e and π
	10.5 Proof of Liouville\'s Theorem
	10.6 Thue\'s Theorem
Appendix  Bibliography
Index