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Algebra

ISBN: 9780130047632 | 0130047635
Edition: 1st
Format: Hardcover
Publisher: Addison Wesley
Pub. Date: 1/1/1991

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SummaryTable of Contents
M->CREATED
This book is for the honors undergraduate or introductory graduate course. Linear algebra is tightly integrated into the text.
... MORE
Prefacexiii
A Note for the Teacherxv
Matrix Operations
1(37)
The Basic Operations
1(8)
Row Reduction
9(9)
Determinants
18(6)
Permutation Matrices
24(4)
Cramer's Rule
28(10)
Exercises
31(7)
Groups
38(40)
The Definition of a Group
38(6)
Subgroups
44(4)
Isomorphisms
48(3)
Homomorphisms
51(2)
Equivalence Relations and Partitions
53(4)
Cosets
57(2)
Restriction of a Homomorphism to a Subgroup
59(2)
Products of Groups
61(3)
Modular Arithmetic
64(2)
Quotient Groups
66(12)
Exercises
69(9)
Vector Spaces
78(31)
Real Vector Spaces
78(4)
Abstract Fields
82(5)
Bases of Dimension
87(7)
Computation with Bases
94(6)
Infinite-Dimensional Spaces
100(2)
Direct Sums
102(7)
Exercises
104(5)
Linear Transformations
109(46)
The Dimension Formula
109(2)
The Matrix of a Linear Transformation
111(4)
Linear Operators and Eigenvectors
115(5)
The Characteristic Polynomial
120(3)
Orthogonal Matrices and Rotations
123(7)
Diagonalization
130(3)
Systems of Differential Equations
133(5)
The Matrix Exponential
138(17)
Exercises
145(10)
Symmetry
155(42)
Symmetry of Plane Figures
155(2)
The Group of Motions of the Plane
157(5)
Finite Groups of Motions
162(4)
Discrete Groups of Motions
166(9)
Abstract Symmetry: Group Operations
175(3)
The Operation on Cosets
178(2)
The Counting Formula
180(2)
Permutation Representations
182(2)
Finite Subgroups of the Rotation Group
184(13)
Exercises
188(9)
More Group Theory
197(40)
The Operations of a Group on Itself
197(3)
The Class Equation of the Icosahedral Group
200(3)
Operations on Subsets
203(2)
The Sylow Theorems
205(4)
The Groups of Order 12
209(2)
Computation in the Symmetric Group
211(6)
The Free Group
217(2)
Generators and Relations
219(4)
The Todd--Coxeter Algorithm
223(14)
Exercises
229(8)
Bilinear Forms
237(33)
Definition of Bilinear Form
237(6)
Symmetric Forms: Orthogonality
243(4)
The Geometry Associated to a Positive Form
247(2)
Hermitian Forms
249(4)
The Spectral Theorem
253(2)
Conics and Quadrics
255(4)
The Spectral Theorem for Normal Operators
259(1)
Skew-Symmetric Forms
260(1)
Summary of Results, in Matrix Notation
261(9)
Exercises
262(8)
Linear Groups
270(37)
The Classical Linear Groups
270(2)
The Special Unitary Group SU2
272(4)
The Orthogonal Representation of SU2
276(5)
The Special Linear Group SL2(R)
281(2)
One-Parameter Subgroups
283(3)
The Lie Algebra
286(6)
Translation in a Group
292(3)
Simple Groups
295(12)
Exercises
300(7)
Group Representations
307(38)
Difinition of a Group Representation
307(3)
G-Invariant Forms and Unitary Representations
310(2)
Compact Groups
312(2)
G-Invariant Subspaces and Irreducible Representations
314(2)
Characters
316(5)
Permutation Representations and the Regular Representation
321(2)
The Reprentations of the Icosahedral Group
323(2)
One-Dimensional Representations
325(1)
Schur's Lemma, and Proof of the Orthogonality Relations
325(5)
Representations of the Group SU2
330(15)
Exercises
335(10)
Rings
345(44)
Definition of a Ring
345(2)
Formal Construction of Integers and Polynomials
347(6)
Homomorphisms and Ideals
353(6)
Quotient Rings and Relations in a Ring
359(5)
Adjunction of Elements
364(4)
Integral Domains and Fraction Fields
368(2)
Maximal Ideals
370(3)
Algebraic Geometry
373(16)
Exercises
379(10)
Factorization
389(61)
Factorization of Integers and Polynomials
389(3)
Unique Factorization Domains, Principal Ideal Domains, and Euclidean Domains
392(6)
Gauss's Lemma
398(4)
Explicit Factorization of Polynomials
402(4)
Primes in the Ring of Gauss Integers
406(3)
Algebraic Integers
409(5)
Factorization in Imaginary Quadratic Fields
414(5)
Ideal Factorization
419(5)
The Relation Between Prime Ideals of R and Prime Integers
424(1)
Ideal Classes in Imaginary Quadratic Fields
425(8)
Real Qudratic Fields
433(4)
Some Diophantine Equations
437(13)
Exercises
440(10)
Modules
450(42)
The Definition of a Module
450(2)
Matrices, Free Modules, and Bases
452(4)
The Principle of Permanence of Identities
456(1)
Diagonalization of Integer Matrices
457(7)
Generators and Relations for Modules
464(7)
The Structure Theorem for Abelian Groups
471(5)
Application to Linear Operators
476(6)
Free Modules over Polynomial Rings
482(10)
Exercises
483(9)
Fields
492(45)
Examples of Fields
492(1)
Algebraic and Transcendental Elements
493(3)
The Degree of a Field Extension
496(4)
Constructions with Ruler and Compass
500(6)
Symbolic Adjunction of Roots
506(3)
Finite Fields
509(6)
Function Fields
515(10)
Transcendental Extensions
525(2)
Algebraically Closed Fields
527(10)
Exercises
530(7)
Galois Theory
537(48)
The Main Theorem of Galois Theory
537(6)
Cubic Equations
543(4)
Symmetric Functions
547(5)
Primitive Elements
552(4)
Proof of the Main Theorem
556(4)
Quartic Equations
560(5)
Kummer Extensions
565(2)
Cyclotomic Extensions
567(3)
Quintic Equations
570(15)
Exercises
575(10)
Appendix Background Material585(16)
1. Set Theory
585(4)
2. Techniques of Proof
589(4)
3. Topology
593(4)
4. The Implicit Function Theorem
597(4)
Exercises
599(2)
Notation601(2)
Suggestions for Further Reading603(4)
Index607

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