CONTENTS
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Preface
Chapter 1 Why Abstract Algebra?
History of Algebra. New Algebras. Algebraic Structures. Axioms and Axiomatic Algebra.
Abstraction in Algebra.
Chapter 2 Operations
Operations on a Set. Properties of Operations.
Chapter 3 The Definition of Groups
Groups. Examples of Infinite and Finite Groups. Examples of Abelian and Nonabelian
Groups. Group Tables.
Theory of Coding: Maximum-Likelihood Decoding.
Chapter 4 Elementary Properties of Groups
Uniqueness of Identity and Inverses. Properties of Inverses.
Direct Product of Groups.
Chapter 5 Subgroups
Definition of Subgroup. Generators and Defining Relations.
Cayley Diagrams. Center of a Group. Group Codes; Hamming Code
.
Chapter 6 Functions
Injective, Surjective, Bijective Function. Composite and Inverse of Functions.
Finite-State Machines. Automata and Their Semigroups.
Chapter 7 Groups of Permutations
Symmetric Groups. Dihedral Groups.
An Application of Groups to Anthropology.
Chapter 8 Permutations of a Finite Set
Decomposition of Permutations into Cycles. Transpositions. Even and Odd Permutations.
Alternating Groups.
Chapter 9 Isomorphism
The Concept of Isomorphism in Mathematics. Isomorphic and Nonisomorphic Groups.
Cayley’s Theorem.
Group Automorphisms
.
Chapter 10 Order of Group Elements
Powers/Multiples of Group Elements. Laws of Exponents. Properties of the Order of Group Elements.
Chapter 11 Cyclic Groups
Finite and Infinite Cyclic Groups. Isomorphism of Cyclic Groups. Subgroups of Cyclic
Groups.
Chapter 12 Partitions and Equivalence Relations
Chapter 13 Counting Cosets
Lagrange’s Theorem and Elementary Consequences.
Survey of Groups of Order ≤ 10.
Number of Conjugate Elements. Group Acting on a Set.
Chapter 14 Homomorphisms
Elementary Properties of Homomorphisms. Normal Subgroups. Kernel and Range.
Inner Direct Products. Conjugate Subgroups.
Chapter 15 Quotient Groups
Quotient Group Construction. Examples and Applications.
The Class Equation. Induction on the Order of a Group.
Chapter 16 The Fundamental Homomorphism Theorem
Fundamental Homomorphism Theorem and Some Consequences.
The Isomorphism Theorems. The Correspondence Theorem. Cauchy’s Theorem. Sylow
Subgroups. Sylow’s Theorem. Decomposition Theorem for Finite Abelian Groups
.
Chapter 17 Rings: Definitions and Elementary Properties
Commutative Rings. Unity. Invertibles and Zero-Divisors. Integral Domain. Field.
Chapter 18 Ideals and Homomorphisms
Chapter 19 Quotient Rings
Construction of Quotient Rings. Examples. Fundamental Homomorphism Theorem and
Some Consequences. Properties of Prime and Maximal Ideals.
Chapter 20 Integral Domains
Characteristic of an Integral Domain. Properties of the Characteristic. Finite Fields.
Construction of the Field of Quotients.
Chapter21 The Integers
Ordered Integral Domains. Well-ordering. Characterization of
Up to Isomorphism.
Mathematical Induction. Division Algorithm.
Chapter 22 Factoring into Primes
Ideals of Z. Properties of the GCD. Relatively Prime Integers. Primes. Euclid’s Lemma.
Unique Factorization.
Chapter 23 Elements of Number Theory (Optional)
Properties of Congruence. Theorems of Fermât and Euler. Solutions of Linear Congruences.
Chinese Remainder Theorem.
Wilson’s Theorem and Consequences. Quadratic Residues. The Legendre Symbol.
Primitive Roots.
Chapter 24 Rings of Polynomials
Motivation and Definitions. Domain of Polynomials over a Field. Division Algorithm.
Polynomials in Several Variables. Fields of Polynomial Quotients.
Chapter 25 Factoring Polynomials
Ideals of F[x]. Properties of the GCD. Irreducible Polynomials. Unique factorization.
Euclidean Algorithm.
Chapter 26 Substitution in Polynomials
Roots and Factors. Polynomial Functions. Polynomials over Q
Eisenstein’s Irreducibility Criterion.
Polynomials over the Reals. Polynomial Interpolation.
Chapter 27 Extensions of Fields
Algebraic and Transcendental Elements. The Minimum Polynomial. Basic Theorem on
Field Extensions.
Chapter 28 Vector Spaces
Elementary Properties of Vector Spaces. Linear Independence. Basis. Dimension. Linear Transformations.
Chapter29 Degrees of Field Extensions
Simple and Iterated Extensions. Degree of an Iterated Extension.
Fields of Algebraic Elements. Algebraic Numbers. Algebraic Closure.
Chapter 30 Ruler and Compass
Constructible Points and Numbers. Impossible Constructions.
Constructible Angles and Polygons.
Chapter 31 Galois Theory: Preamble
Multiple Roots. Root Field. Extension of a Field. Isomorphism.
Roots of Unity. Separable Polynomials. Normal Extensions.
Chapter 32 Galois Theory: The Heart of the Matter
Field Automorphisms. The Galois Group. The Galois Correspondence. Fundamental
Theorem of Galois Theory.
Computing Galois Groups.
Chapter 33
Solving Equations by Radicals
Radical Extensions. Abelian Extensions. Solvable Groups. Insolvability of the Quin tic.
Appendix A Review of Set Theory
Appendix B Review of the Integers
Appendix C Review of Mathematical Induction
Answers to Selected Exercises
Index
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