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Basic Set, Mapping, and Logic Notation
Symbols
Title
Preface
Introduction
Basic Properties of the Integers
Mathematical Induction
Multiples and Divisors, Primes in Z
The Division Algorithm
Common Divisors
Euclid’s Algorithm
Common Multiples
Unique Factorization in Z
Groups
Permutations on a Finite Set
Multiplication of Permutations
Abstract Groups
Cycle Notation
Subgroups in a Group
Additive Notation, Modular Arithmetic
Cycllc Groups
Even and Odd Permutations
Groups of Symmetries
The Alternating Groups An
Cosets of a Subgroup
Quotient Groups
Solvable Groups
More on Symmetry
The Sylow Theorems (Without Proofs)
Sets and Mappings
Mappings
Group Isomorphisms
Group Homomorphisms
Cayley's Theorem
Unions, Intersections, Partitions
Cartesian Products, Direct Products
Relations
Partially Ordered Sets
Power Sets
Operations
Algebraic Structures
Boolean Algebra
Boolean Functions and Their Applications
Composition of Mappings, Groups of Bijections
Conjugacy Classes and the Class Equation
The Fundamental Theorem on Abelian Groups (Without Proof)
Rings and Fields
Rings
Ring Homomorphisms and Ideals
Congruence in Z, The Euler and Fermat Theorems
lntegral Domains
Fields
Ordered Integral Domains and Fields
Matrices, Quaternions
Embedding a Ring in a Field
Characterizations of Z, Q, R, and C
Polynomials
Polynomial Extensions of Rings
Polynomials over a Commutative Ring
Divisibility in Commutative Rings
Polynomial Functions
Integral and Rational Roots
Polynomial Fitting, Finite Differences
Ideals in F[x]
Factorizations of Polynomials
Partial Fractions
Extension Fields
Equations of Degree 2, 3, or 4
Automorphisms of E over F, lnsolvability of a Quintic
Euclidean Constructions
Closure Under Euclidean Constructions
Closure Under Square Roots
Constructible Points, Lines, and Circles
More on the Integers
Simultaneous Congruences -- The Chinese Remainder Theorem
More on Mathematical Induction
Some Number Theoretic Functions
Quadratic Residues
Some Applications to Coding
Binary Codes
Matrix Codes, A Hamming Code
Modular Codes, Trapdoor Functions
Computer Programming Projects
Supplementary and Challenging Problems
Annotated Bibliography
Answers, Hints, or Solutions for Most Odd-Numbered Problems
Index
Summary of Axioms for Groups, Rings, and Fields
Some Group Tables