Pinter C.C. A Book of Abstract Algebra

McGraw-Hill Higher Education, 1982. - 350 Pages.This text is aimed at the abstract or modern algebra course taken by junior and senior math majors and many secondary math education majors. A mid-level approach, this text features clear prose, an intuitive approach, and exercises organized around specific concepts. New to this edition are additional applications exercises to improve student learning.Accessible but rigorous, this outstanding text encompasses all of the topics covered by a typical course in elementary abstract algebra. Its easy-to-read treatment offers an intuitive approach, featuring informal discussions followed by thematically arranged exercises. This second edition features additional exercises to improve student familiarity with applications.Why Abstract Algebra?
Origins
The Algebra of Matrices
Boolean Algebra
Algebraic Structures
Axioms And Men
The Axiomatics Of Algebra
Abstraction Revisited
Operations
Properties of Operations
Operations on a Two-Element Set
The Definition Of Groups
Examples of Abelian Groups
Groups of Subsets of a Set
A Checkerboard Game
A Coin Game
Groups in Binary Codes
Elementary Properties Of Groups
Solving Equations in Groups
Group Elements and Their Inverses
Counting Elements and Their Inverses
Constructing Small Groups
Powers and Roots of Group Elements
Subgroups
Recognizing Subgroups
Subgroups of Abelian Groups
Generators of Groups
Cayley Diagrams
Functions
Examples of Injective and Surjective Functions
Composite Functions
Inverses of Functions
Some General Properties of Functions
Groups Of Permutations
Computing Elements of S6
Examples of Groups of Permutatiotis
Symmetries of Geometric Figures
Symmetries of Polynomials
Properties of Permutations of a Set A
Permutations Of A Finite Set
Practice in Multiplying and Factoring Permutations
Powers of Permutations
Conjugate Cycles
Even/Odd Permutations in Subgroups of Sn
Generators of An and Sn
Isomorphism
Elements Which Correspond under an Isomorphism
Separating Groups into Isomorphism Classes
Isomorphism of Groups Given by Generators and Defining Equations
Group Automorphisms
Regular Representation of Groups
Order Of Group Elements
Laws of Exponents
Elementary Properties of Order
Orders of Powers of Elements
Relationship between the Order of a and the Order of any kthRoot of a
Cyclic Groups
Generators of Cyclic Groups
Direct Products of Cyclic Groups
kth Roots of Elements in a Cyclic Group
Partitions And Equivalence Relations
Examples of Equivalence Relations
General Properties of Equivalence Relations and Partitions
Counting Cosets
Examples of Cosets in Infinite groups
Elementary Properties of Cosets
Survey of Al! 10-Element Groups
Conjugate Elements
Group Acting on a Set
Homomorphisms
Examples of Homomorphisms of Finite Groups
Elementary Properties of Homomorphisms
Homomorphism and the Order of Elements
Conjugate Subgroups
Quotient Groups
Examples of Finite Quotient Groups
Relating Properties of H to Properties of G/H
Quotient of a Group by its Center
Using the Class Equation to Determine the Size of the Center
Induction on IG| : An Example
The Fundamental Homomorphism Theorem
Examples of the FHT Applied to Finite Groups
Group of Inner Automorphisms of a Group G
The First Isomorphism Theorem
Quotient Groups Isomorphic to the Circle Group
Canchy 's Theorem
p-Sylow Subgroups
Sylow's Theorem
Decomposition of a Finite Abelian Group into p-Groups
Basis Theorem for Finite Abelian Groups
Rings: Definitions And Elementary Properties
Ring of 2 x 2 Matrices
Ring of Quaternions
Ring of Endomorphisms
Properties of Invertible Elements
The Binomial Formula
Nilpotent and Unipotent Elements
Ideals And Homomorphisms
Examples of Subrings
Elementary Properties of Subrings
Elementary Properties of Homomorphisms
Further Properties of Ideals
A Ring of Endomorphisms
Quotient Rings
Examples of Quotient Rings
Quotient Rings and Homomorphic Images in F(R)
Properties of Quotient Rings A/J in Relation to Properties of J
Zn as a Homomorphic Image of Z
Integral Domains
Optional
Characteristic of a Finite Integral Domain
Further Properties of the Characteristic of an Integral Domain
Finite Fields
The Integers
Uses of Induction
Absolute Values
Principle of Strong Induction
Factoring Into Primes
Properties of the ged
Least Common Multiples
The gcd and the 1cm as Operations on /
Elements Of Number Theory
Optional
Solving Single Congruences
Elementary Properties of Congruence
Consequences of Fermat's Theorem
Wilson's Theorem, and Some Consequences
Quadratic Residues
Primitive Roots
Rings Of Polynomials
Problems Involving Concepts and Definitions
Domains A[x] where A Has Finite Characteristic
Homomorphisms of Domains of Polynomials
Fields of Polynomial Quotients
Division Algorithm: Uniqueness of Quotient and Remainder
Factoring Polynomials
Short Questions Relating to Irreducible Polynomials
A Method for Computing the gcd
An Automorphism of F[x]
Substitution In Polynomials
Polynomials Over Y And Q
Polynomials Over R And C
Finding Roots of Polynomials over Q
Irreducible Polynomials in Q[x] by Eisenstein 's Criterion (and Variations on the Theme)
Mapping onto Zn, to Determine Irreducibility over Q
Polynomial Functions over a Finite Field
Polynomial Interpolation
Extensions Of Fields
Recognizing Algebraic Elements
The Structure of Fields F[x]
Simple Extensions
Questions Relating to Transcendental Elements
Multiple Roots
Vector Spaces
Examples of Subspaces
Properties of Linear Transformations
Sums of Vector Spaces
Degrees Of Field Extensions
Examples of Finite Extensions
Degrees of Extensions (Applications of Theorem 2)
Further Properties of Degrees of Extensions
Fields of Algebraic Elements: Algebraic Numbers
Ruler And Compass
Constructible Numbers
Constructible Polygons
A Nonconstructible Polygon
Further Properties of Constructible Numbers and Figures
Galois Theory: Preamble
Examples of Root Fields over Zp
Reducing Iterated Extensions to Simple Extensions
Separable and Inseparable Polynomials
Multiple Roots over Infinite Fields of Nonzero Characteristic
Extending Isomorphisms
Normal Extensions
Galois Theory: The Heart Ofthe Matter
Computing a Galois Group
A Galois Group Equal to D4
A Galois Group Isomorphic to S5
Further Questions Relating to Galois Groups
Normal Extensions and Normal Subgroups
Solving Equations By Radicals
Finding Radical Extensions
pth Roots of Elements in a Field
If GaI(K: F) Is Solvable, K Is a Radical Extension of F