9781584885153

A First Course in Abstract Algebra: Rings, Groups and Fields, Second Edition

by ;
  • ISBN13:

    9781584885153

  • ISBN10:

    1584885157

  • Edition: 2nd
  • Format: Hardcover
  • Copyright: 2005-01-27
  • Publisher: Chapman & Hall/

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Summary

Most abstract algebra texts begin with groups, then proceed to rings and fields. While groups are the logically simplest of the structures, the motivation for studying groups can be somewhat lost on students approaching abstract algebra for the first time. To engage and motivate them, starting with something students know and abstracting from there is more natural-and ultimately more effective.Authors Anderson and Feil developed A First Course in Abstract Algebra: Rings, Groups and Fields based upon that conviction. The text begins with ring theory, building upon students' familiarity with integers and polynomials. Later, when students have become more experienced, it introduces groups. The last section of the book develops Galois Theory with the goal of showing the impossibility of solving the quintic with radicals.Each section of the book ends with a "Section in a Nutshell" synopsis of important definitions and theorems. Each chapter includes "Quick Exercises" that reinforce the topic addressed and are designed to be worked as the text is read. Problem sets at the end of each chapter begin with "Warm-Up Exercises" that test fundamental comprehension, followed by regular exercises, both computational and "supply the proof" problems. A Hints and Answers section is provided at the end of the book.As stated in the title, this book is designed for a first course--either one or two semesters in abstract algebra. It requires only a typical calculus sequence as a prerequisite and does not assume any familiarity with linear algebra or complex numbers.

Table of Contents

Preface xiii
I Numbers, Polynomials, and Factoring
The Natural Numbers
3(12)
Operations on the Natural Numbers
3(1)
Well Ordering and Mathematical Induction
4(3)
The Fibonacci Sequence
7(2)
Well Ordering Implies Mathematical Induction
9(1)
The Axiomatic Method
9(6)
The Integers
15(16)
The Division Theorem
15(2)
The Greatest Common Divisor
17(3)
The GCD Identity
20(2)
The Fundamental Theorem of Arithmetic
22(2)
A Geometric Interpretation
24(7)
Modular Arithmetic
31(10)
Residue Classes
31(3)
Arithmetic on the Residue Classes
34(2)
Properties of Modular Arithmetic
36(5)
Polynomials with Rational Coefficients
41(14)
Polynomials
41(2)
The Algebra of Polynomials
43(2)
The Analogy between Z and Q[x]
45(2)
Factors of a Polynomial
47(1)
Linear Factors
47(2)
Greatest Common Divisors
49(6)
Factorization of Polynomials
55(18)
Factoring Polynomials
55(2)
Unique Factorization
57(2)
Polynomials with Integer Coefficients
59(14)
Section I in a Nutshell
69(4)
II Rings, Domains, and Fields
Rings
73(16)
Binary Operations
73(1)
Rings
74(6)
Arithmetic in a Ring
80(1)
Notational Conventions
81(1)
The Set of Integers is a Ring
82(7)
Subrings and Unity
89(12)
Subrings
89(4)
The Multiplicative Identity
93(8)
Integral Domains and Fields
101(18)
Zero Divisors
101(2)
Units
103(2)
Fields
105(1)
The Field of Complex Numbers
105(5)
Finite Fields
110(9)
Polynomials over a Field
119(22)
Polynomials with Coefficients from an Arbitrary Field
119(2)
Polynomials with Complex Coefficients
121(3)
Irreducibles in R[x]
124(1)
Extraction of Square Roots in C
125(16)
Section II in a Nutshell
135(6)
III Unique Factorization
Associates and Irreducibles
141(14)
Associates
141(1)
Irreducibles
142(2)
Quadratic Extensions of the Integers
144(1)
Units in Quadratic Extensions
145(4)
Irreducibles in Quadratic Extensions
149(6)
Factorization and Ideals
155(14)
Factorization for Quadratic Extensions
155(2)
How Might Factorization Fail?
157(1)
Ideals
158(2)
Principal Ideals
160(9)
Principal Ideal Domains
169(8)
Ideals that are not Principal
169(2)
Principal Ideal Domains
171(6)
Primes and Unique Factorization
177(12)
Primes
177(2)
UFDs
179(1)
Expressing Properties of Elements in Terms of Ideals
180(3)
Ideals in Z [√-5]
183(1)
A Comparison between Z and Z [√-5]
183(1)
All PIDs are UFDs
184(5)
Polynomials with Integer Coefficients
189(8)
The Proof that Q[x] is a UFD
189(1)
Factoring Integers out of Polynomials
190(1)
The Content of a Polynomial
191(2)
Irreducibles in Z[x] are Prime
193(4)
Euclidean Domains
197(14)
Euclidean Domains
197(2)
The Gaussian Integers
199(2)
Euclidean Domains are PIDs
201(2)
Some PIDs are not Euclidean
203(8)
Section III in a Nutshell
207(4)
IV Ring Homomorphisms and Ideals
Ring Homomorphisms
211(14)
Homomorphisms
211(3)
One-to-one and Onto Functions
214(1)
Properties Preserved by Homomorphisms
215(1)
More Examples
216(2)
Making a Homomorphism Onto
218(7)
The Kernel
225(12)
Ideals
226(1)
The Kernel
226(2)
The Kernel is an Ideal
228(1)
All Pre-images Can Be Obtained from the Kernel
229(3)
When is the Kernel Trivial?
232(1)
A Summary and Example
232(5)
Rings of Cosets
237(10)
The Ring of Cosets
237(3)
The Natural Homomorphism
240(7)
The Isomorphism Theorem for Rings
247(12)
Isomorphism
247(2)
The Fundamental Isomorphism Theorem
249(2)
Examples
251(8)
Maximal and Prime Ideals
259(12)
Maximal Ideals
259(3)
Prime Ideals
262(9)
The Chinese Remainder Theorem
271(16)
Direct Products of Domains
271(3)
Chinese Remainder Theorem
274(13)
Section IV in a Nutshell
283(4)
V Groups
Symmetries of Figures in the Plane
287(12)
Symmetries of the Equilateral Triangle
287(3)
Permutation Notation
290(2)
Matrix Notation
292(2)
Symmetries of the Square
294(5)
Symmetries of Figures in Space
299(14)
Symmetries of the Regular Tetrahedron
300(4)
Symmetries of the Cube
304(9)
Abstract Groups
313(16)
Definition of Group
314(1)
Examples of Groups
314(2)
Multiplicative Groups
316(13)
Subgroups
329(10)
Arithmetic in an Abstract Group
329(1)
Notation
330(1)
Subgroups
331(2)
Characterization of Subgroups
333(6)
Cyclic Groups
339(18)
The Order of an Element
339(3)
Rule of Exponents
342(3)
Cyclic Subgroups
345(2)
Cyclic Groups
347(10)
Section V in a Nutshell
353(4)
VI Group Homomorphisms and Permutations
Group Homomorphisms
357(10)
Homomorphisms
357(1)
Examples
358(3)
Direct Products
361(6)
Group Isomorphisms
367(12)
Structure Preserved by Homomorphisms
368(1)
Uniqueness of Cyclic Groups
369(2)
Symmetry Groups
371(1)
Characterizing Direct Products
372(7)
Permutations and Cayley's Theorem
379(10)
Permutations
379(1)
The Symmetric Groups
380(3)
Cayley's Theorem
383(6)
More About Permutations
389(10)
Cycles
389(2)
Cycle Factorization of Permutations
391(3)
Orders of Permutations
394(5)
Cosets and Lagrange's Theorem
399(14)
Cosets
399(2)
Lagrange's Theorem
401(3)
Applications of Lagrange's Theorem
404(9)
Groups of Cosets
413(12)
Left Cosets
414(1)
Normal Subgroups
415(2)
Examples of Groups of Cosets
417(8)
The Isomorphism Theorem for Groups
425(10)
The Kernel
425(3)
Cosets of the Kernel
428(1)
The Fundamental Theorem
429(6)
The Alternating Groups
435(14)
Transpositions
435(1)
The Parity of a Permutation
436(2)
The Alternating Groups
438(1)
The Alternating Subgroup is Normal
439(3)
Simple Groups
442(7)
Fundamental Theorem for Finite Abelian Groups
449(6)
The Fundamental Theorem
449(3)
p-groups
452(3)
Solvable Groups
455(10)
Solvability
455(2)
New Solvable Groups from Old
457(8)
Section VI in a Nutshell
461(4)
VII Constructibility Problems
Constructions with Compass and Straightedge
465(10)
Construction Problems
465(2)
Constructible Lengths and Numbers
467(8)
Constructibility and Quadratic Field Extensions
475(14)
Quadratic Field Extensions
475(2)
Sequences of Quadratic Field Extensions
477(2)
The Rational Plane
479(1)
Planes of Constructible Numbers
480(4)
The Constructible Number Theorem
484(5)
The Impossibility of Certain Constructions
489(14)
Doubling the Cube
489(1)
Trisecting the Angle
490(3)
Squaring the Circle
493(10)
Section VII in a Nutshell
499(4)
VIII Vector Spaces and Field Extensions
Vector Spaces I
503(8)
Vectors
504(1)
Vector Spaces
505(6)
Vector Spaces II
511(16)
Spanning Sets
511(3)
A Basis for a Vector Space
514(4)
Finding a Basis
518(2)
Dimension of a Vector Space
520(7)
Field Extensions and Kronecker's Theorem
527(10)
Field Extensions
527(1)
Kronecker's Theorem
528(2)
The Characteristic of a Field
530(7)
Algebraic Field Extensions
537(14)
The Minimal Polynomial for an Element
537(2)
Simple Extensions
539(5)
Simple Transcendental Extensions
544(1)
Dimension of Simple Algebraic Extensions
545(6)
Finite Extensions and Constructibility Revisited
551(14)
Finite Extensions
551(5)
Constructibility Problems
556(9)
Section VIII in a Nutshell
561(4)
IX Galois Theory
The Splitting Field
565(12)
The Splitting Field
566(4)
Fields with Characteristic Zero
570(7)
Finite Fields
577(8)
Existence and Uniqueness
577(3)
Examples
580(5)
Galois Groups
585(14)
The Galois Group
585(3)
Galois Groups of Splitting Fields
588(11)
The Fundamental Theorem of Galois Theory
599(16)
Subgroups and Subfields
599(2)
Symmetric Polynomials
601(1)
The Fixed Field and Normal Extensions
602(2)
The Fundamental Theorem
604(3)
Examples
607(8)
Solving Polynomials by Radicals
615(18)
Field Extensions by Radicals
615(3)
Refining the Root Tower
618(4)
Solvable Galois Groups
622(11)
Section IX in a Nutshell
629(4)
Hints and Solutions 633(28)
Guide to Notation 661(4)
Index 665

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