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Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
|---|---|---|---|---|---|
Searching... | 30000010344129 | QA162 F56 2014 | Open Access Book | Book | Searching... |
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Summary
Summary
A new approach to abstract algebra that eases student anxieties by building on fundamentals.
Introduction to Abstract Algebra presents a breakthrough approach to teaching one of math's most intimidating concepts. Avoiding the pitfalls common in the standard textbooks, Benjamin Fine, Anthony M. Gaglione, and Gerhard Rosenberger set a pace that allows beginner-level students to follow the progression from familiar topics such as rings, numbers, and groups to more difficult concepts.
Classroom tested and revised until students achieved consistent, positive results, this textbook is designed to keep students focused as they learn complex topics. Fine, Gaglione, and Rosenberger's clear explanations prevent students from getting lost as they move deeper and deeper into areas such as abelian groups, fields, and Galois theory.
This textbook will help bring about the day when abstract algebra no longer creates intense anxiety but instead challenges students to fully grasp the meaning and power of the approach.
Topics covered include:
* Rings
* Integral domains
* The fundamental theorem of arithmetic
* Fields
* Groups
* Lagrange's theorem
* Isomorphism theorems for groups
* Fundamental theorem of finite abelian groups
* The simplicity of A n for n5
* Sylow theorems
* The Jordan-Hölder theorem
* Ring isomorphism theorems
* Euclidean domains
* Principal ideal domains
* The fundamental theorem of algebra
* Vector spaces
* Algebras
* Field extensions: algebraic and transcendental
* The fundamental theorem of Galois theory
* The insolvability of the quintic
Author Notes
Benjamin Fine is a professor of mathematics at Fairfield University. Anthony M. Gaglione is a professor of mathematics at the United States Naval Academy. Gerhard Rosenberger is a professor of mathematics at the University of Hamburg, Germany.
Table of Contents
| 0 Preface |
| 1 Abstract Algebra and Algebraic Reasoning |
| 1.1 Abstract Algebra |
| 1.2 Algebraic Structures |
| 1.3 The\Algebraic Method |
| 1.4 The\Standard Number Systems |
| 1.5 The\Integers and Induction |
| 1.6 Exercises |
| 2 Algebraic Preliminaries |
| 2.1 Sets and Set Theory |
| 2.1.1 Set Operations |
| 2.2 Functions |
| 2.3 Equivalence Relations and Factor Sets |
| 2.4 Sizes of Sets |
| 2.5 Binary Operations |
| 2.5.1 The\Algebra of Sets |
| 2.6 Algebraic Structures and Isomorphisms |
| 2.7 Groups |
| 2.8 Exercises |
| 3 Rings and the Integers |
| 3.1 Rings and the Ring of Integers |
| 3.2 Some Basic Properties of Rings and Subrings |
| 3.3 Examples of Rings |
| 3.3.1 The\Modular Rings |
| 3.3.2 Noncommutative Rings |
| 3.3.3 Rings Without Identities |
| 3.3.4 Rings of Subsets |
| 3.3.5 Direct Sums of Rings |
| 3.3.6 Summary of Examples |
| 3.4 Ring Homomorphisms and Isomorphisms |
| 3.5 Integral Domains and Ordering |
| 3.6 Mathematical Induction and the Uniqueness of Z |
| 3.7 Exercises |
| 4 Number Theory and Unique Factorization |
| 4.1 Elementary Number Theory |
| 4.2 Divisibility and Primes |
| 4.3 Greatest Common Divisors |
| 4.4 The\Fundamental Theorem of Arithmetic |
| 4.5 Congruences and Modular Arithmetic |
| 4.6 Unique Factorization Domains |
| 4.7 Exercises |
| 5 Fields |
| 5.1 Fields and Division Rings |
| 5.2 Construction and Uniqueness of the Rationals |
| 5.2.1 Fields of Fractions |
| 5.3 The\Real Number System |
| 5.3.1 The\Completeness of R (Optional) |
| 5.3.2 Characterization of R (Optional) |
| 5.3.3 The\Construction of R (Optional) |
| 5.3.4 The\p-adic Numbers (Optional) |
| 5.4 The\Field of Complex Numbers |
| 5.4.1 Geometric Interpretation |
| 5.4.2 Polar Form and Euler's Identity |
| 5.4.3 DeMoivre's Theorem for Powers and Roots |
| 5.5 Exercises |
| 6 Basic Group Theory |
| 6.1 Groups, Subgroups and Isomorphisms |
| 6.2 Examples of Groups |
| 6.2.1 Permutations and the Symmetric Group |
| 6.2.2 Examples of Groups |
| 6.3 Subgroups and Lagrange's Theorem |
| 6.4 Generators and Cyclic Groups |
| 6.5 Exercises |
| 7 Factor Groups and the Group Isomorphism Theorems |
| 7.1 Normal Subgroups |
| 7.2 Factor Groups |
| 7.2.1 Examples of Factor Groups |
| 7.3 The\Group Isomorphism Theorems |
| 7.4 Exercises |
| 8 Direct Products and Abelian Groups |
| 8.1 Direct Products of Groups |
| 8.1.1 Direct Products of Two Groups |
| 8.1.2 Direct Products of Any Finite Number of Groups |
| 8.2 Abelian Groups |
| 8.2.1 Finite Abelian Groups |
| 8.2.2 Free Abelian Groups |
| 8.2.3 The\Basis Theorem for Finitely Generated Abelian Groups |
| 8.3 Exercises |
| 9 Symmetric and Alternating Groups |
| 9.1 Symmetric Groups and Cycle Structure |
| 9.1.1 The\Alternating Groups |
| 9.1.2 Conjugation in Sn |
| 9.2 The\Simplicity of An |
| 9.3 Exercises |
| 10 Group Actions and Topics in Group Theory |
| 10.1 Group Actions |
| 10.2 Conjugacy Classes and the Class Equation |
| 10.3 The\Sylow Theorems |
| 10.3.1 Some Applications of the Sylow Theorems |
| 10.4 Groups of Small Order |
| 10.5 Solvability and Solvable Groups |
| 10.5.1 Solvable Groups |
| 10.5.2 The\Derived Series |
| 10.6 Composition Series and the Jordan-Holder Theorem |
| 10.7 Exercises |
| 11 Topics in Ring Theory |
| 11.1 Ideals in Rings |
| 11.2 Factor Rings and the Ring Isomorphism Theorem |
| 11.3 Prime and Maximal Ideals |
| 11.3.1 Prime Ideals and Integral Domains |
| 11.3.2 Maximal Ideals and Fields |
| 11.4 Principal Ideal Domains and Unique Factorization |
| 11.5 Exercises |
| 12 Polynomials and Polynomial Rings |
| 12.1 Polynomials and Polynomial Rings |
| 12.2 Polynomial Rings over a Field |
| 12.2.1 Unique Factorization of Polynomials |
| 12.2.2 Euclidean Domains |
| 12.2.3 F[x] as a Principal Ideal Domain |
| 12.2.4 Polynomial Rings over Integral Domains |
| 12.3 Zeros of Polynomials |
| 12.3.1 Real and Complex Polynomials |
| 12.3.2 The\Fundamental Theorem of Algebra |
| 12.3.3 The\Rational Roots Theorem |
| 12.3.4 Solvability by Radicals |
| 12.3.5 Algebraic and Transcendental Numbers |
| 12.4 Unique Factorization in Z[x] |
| 12.5 Exercises |
| 13 Algebraic Linear Algebra |
| 13.1 Linear Algebra |
| 13.1.1 Vector Analysis in R3 |
| 13.1.2 Matrices and Matrix Algebra |
| 13.1.3 Systems of Linear Equations |
| 13.1.4 Determinants |
| 13.2 Vector Spaces over a Field |
| 13.2.1 Euclidean n-Space |
| 13.2.2 Vector Spaces |
| 13.2.3 Subspaces |
| 13.2.4 Bases and Dimension |
| 13.2.5 Testing for Bases in Fn |
| 13.3 Dimension and Subspaces |
| 13.4 Algebras |
| 13.5 Inner Product Spaces |
| 13.5.1 Banach and Hilbert Spaces |
| 13.5.2 The\Gram-Schmidt Process and Orthonormal Bases |
| 13.5.3 The\Closest Vector Theorem |
| 13.5.4 Least-Squares Approximation |
| 13.6 Linear Transformations and Matrices |
| 13.6.1 Matrix of a Linear Transformation |
| 13.6.2 Linear Operators and Linear Functionals |
| 13.7 Exercises |
| 14 Fields and Field Extensions |
| 14.1 Abstract Algebra and Galois Theory |
| 14.2 Field Extensions |
| 14.3 Algebraic Field Extensions |
| 14.4 F-automorphisms, Conjugates and Algebraic Closures |
| 14.5 Adjoining Roots to Fields |
| 14.6 Splitting Fields and Algebraic Closures |
| 14.7 Automorphisms and Fixed Fields |
| 14.8 Finite Fields |
| 14.9 Transcendental Extensions |
| 14.10 Exercises |
| 15 A\Survey of Galois Theory |
| 15.1 An\Overview of Galois Theory |
| 15.2 Galois Extensions |
| 15.3 Automorphisms and the Galois Group |
| 15.4 The\Fundamental Theorem of Galois Theory |
| 15.5 A\Proof of the Fundamental Theorem of Algebra |
| 15.6 Some Applications of Galois Theory |
| 15.6.1 The\Insolvability of the Quintic |
| 15.6.2 Some Ruler and Compass Constructions |
| 15.6.3 Algebraic Extensions of R |
| 15.7 Exercises |
| 0 Bibliography |
| 0 Index |
