abstract algebra 3e, john beachy, william blair [pdf]
Abstract Algebra (3rd Edition) by John A. Beachy, William D. Blair
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Abstract Algebra (3rd Edition) written by
John A. Beachy, Northern Illinois University and
William D. Blair, Northern Illinois University.
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.
Abstract Algebra (3rd Edition) written by
John A. Beachy and
William D. Blair
cover the following topics.
Preface
PREFACE TO THE SECOND EDITION
TO THE STUDENT
WRITING PROOFS
HISTORICAL BACKGROUND
1. INTEGERS
1.1 Divisors
1.2 Primes
1.3 Congruences
1.4 Integers Modulo n .
Notes
2. FUNCTIONS
2.1 Functions .
2.2 Equivalence Relations
2.3 Permutations
Notes
3. GROUPS
3.1 Definition of a Group
3.2 Subgroups
3.3 Constructing Examples
3.4 Isomorphisms
3.5 Cyclic Groups
3.6 Permutation Groups .
3.7 Homomorphisms
3.8 Cosets, Normal Subgroups, and Factor Groups .
Notes
IV Contents
4. POLYNOMIALS
4. 1 Fields; Roots of Polynomials
4.2 Factors .
4.3 Existence of Roots
4.4 Polynomials over Z, Q, R, and C
Notes
5. COMMUTATIV E RINGS
5.1 Commutative Rings ; Integral Domains .
5.2 Ring Homomorphisms
5.3 Ideals and Factor Rings .
5.4 Quotient Fields .
Notes
6. FIELDS
6.1 Algebraic Elements .
6.2 Finite and Algebraic Extensions
6.3 Geometric Constructions
6.4 Splitting Fields .
6.5 Finite Fields
6.6 Irreducible Polynomials over Finite Fields .
6.7 Quadratic Reciprocity
Notes .
7. STRUCTURE OF GROUPS
7.1 Isomorphism Theorems; Automorphisms
7.2 Conjugacy
7.3 Groups Acting on Sets
7.4 The Sylow Theorems
7.5 Finite Abelian Groups
7.6 Solvable Groups
7.7 Simple Groups
8. GALOIS THEORY
8.1 The Galois Group of a Polynomial .
8.2 Multiplicity of Roots
8.3 The Fundamental Theorem of Galois Theory
8.4 Solvability by Radicals .
8.5 Cyclotomic Polynomials
8.6 Computing Galois Groups
9. UNIQUE FACTORIZATION
9.1 Principal Ideal Domains .
9.2 Unique Factorization Domains
9.3 Some Diophantine Equations
Appendix
A.1 Sets .
A.2 Construction of the Number Systems .
A.3 Basic Properties of the Integers
A.4 Induction .
A.5 Complex Numbers
A.6 Solution of Cubic and Quartic Equations
A.7 Dimension of a Vector Space
Bibliography
Selected Answers
INDEX OF SYMBOLS
Index
Open
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