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Highly regarded by instructors in past editions for its sequencing of topics as well as its concrete approach, slightly slower beginning pace, and extensive set of exercises, the latest edition of Abstract Algebra extends the thrust of the widely used earlier editions as it introduces modern abstract concepts only after a careful study of important examples. Beachy and Blair's clear narrative presentation responds to the needs of inexperienced students who stumble over proof writing, who understand definitions and theorems but cannot do the problems, and who want more examples that tie into their previous experience. The authors introduce chapters by indicating why the material is important and, at the same time, relating the new material to things from the student's background and linking the subject matter of the chapter to the broader picture.
Instructors will find the latest edition pitched at a suitable level of difficulty and will appreciate its gradual increase in the level of sophistication as the student progresses through the book. Rather than inserting superficial applications at the expense of important mathematical concepts, the Beachy and Blair solid, well-organized treatment motivates the subject with concrete problems from areas that students have previously encountered, namely, the integers and polynomials over the real numbers.
1. Integers
Divisors
Primes
Congruences
Integers Modulo n
2. Functions
Functions
Equivalence Relations
Permutations
3. Groups
Definition of a Group
Subgroups
Constructing Examples
Isomorphisms
Cyclic Groups
Permutation Groups
Homomorphisms
Cosets, Normal Subgroups, and Factor Groups
4. Polynomials
Fields; Roots of Polynomials
Factors
Existence of Roots
Polynomials over Z, Q, R, and C
5. Commutative Rings
Commutative Rings; Integral Domains
Ring Homomorphisms
Ideals and Factor Rings
Quotient Fields
6. Fields
Algebraic Elements
Finite and Algebraic Extensions
Geometric Constructions
Splitting Fields
Finite Fields
Irreducible Polynomials over Finite Fields
Quadratic Reciprocity
7. Structure of Groups
Isomorphism Theorems; Automorphisms
Conjugacy
Groups Acting on Sets
The Sylow Theorems
Finite Abelian Groups
Solvable Groups
Simple Groups
8. Galois Theory
The Galois Group of a Polynomial
Multiplicity of Roots
The Fundamental Theorem of Galois Theory
Solvability by Radicals
Cyclotomic Polynomials
Computing Galois Groups
9. Unique Factorization
Principal Ideal Domains
Unique Factorization Domains
Some Diophantine Equations
Appendix
Sets
Construction of the Number Systems
Basic Properties of the Integers
Induction
Complex Numbers
Solution of Cubic and Quartic Equations
Dimension of a Vector Space
John A. Beachy and William D. Blair
ISBN13: 978-1577664437Highly regarded by instructors in past editions for its sequencing of topics as well as its concrete approach, slightly slower beginning pace, and extensive set of exercises, the latest edition of Abstract Algebra extends the thrust of the widely used earlier editions as it introduces modern abstract concepts only after a careful study of important examples. Beachy and Blair's clear narrative presentation responds to the needs of inexperienced students who stumble over proof writing, who understand definitions and theorems but cannot do the problems, and who want more examples that tie into their previous experience. The authors introduce chapters by indicating why the material is important and, at the same time, relating the new material to things from the student's background and linking the subject matter of the chapter to the broader picture.
Instructors will find the latest edition pitched at a suitable level of difficulty and will appreciate its gradual increase in the level of sophistication as the student progresses through the book. Rather than inserting superficial applications at the expense of important mathematical concepts, the Beachy and Blair solid, well-organized treatment motivates the subject with concrete problems from areas that students have previously encountered, namely, the integers and polynomials over the real numbers.
Table of Contents
1. Integers
Divisors
Primes
Congruences
Integers Modulo n
2. Functions
Functions
Equivalence Relations
Permutations
3. Groups
Definition of a Group
Subgroups
Constructing Examples
Isomorphisms
Cyclic Groups
Permutation Groups
Homomorphisms
Cosets, Normal Subgroups, and Factor Groups
4. Polynomials
Fields; Roots of Polynomials
Factors
Existence of Roots
Polynomials over Z, Q, R, and C
5. Commutative Rings
Commutative Rings; Integral Domains
Ring Homomorphisms
Ideals and Factor Rings
Quotient Fields
6. Fields
Algebraic Elements
Finite and Algebraic Extensions
Geometric Constructions
Splitting Fields
Finite Fields
Irreducible Polynomials over Finite Fields
Quadratic Reciprocity
7. Structure of Groups
Isomorphism Theorems; Automorphisms
Conjugacy
Groups Acting on Sets
The Sylow Theorems
Finite Abelian Groups
Solvable Groups
Simple Groups
8. Galois Theory
The Galois Group of a Polynomial
Multiplicity of Roots
The Fundamental Theorem of Galois Theory
Solvability by Radicals
Cyclotomic Polynomials
Computing Galois Groups
9. Unique Factorization
Principal Ideal Domains
Unique Factorization Domains
Some Diophantine Equations
Appendix
Sets
Construction of the Number Systems
Basic Properties of the Integers
Induction
Complex Numbers
Solution of Cubic and Quartic Equations
Dimension of a Vector Space