Check nearby libraries
Buy this book
![Loading indicator](/images/ajax-loader-bar.gif)
Last edited by Tom Morris
December 29, 2022 | History
This edition doesn't have a description yet. Can you add one?
Check nearby libraries
Buy this book
![Loading indicator](/images/ajax-loader-bar.gif)
Previews available in: English
Showing 4 featured editions. View all 4 editions?
Edition | Availability |
---|---|
1 |
aaaa
Libraries near you:
WorldCat
|
2 |
eeee
|
3 |
eeee
Libraries near you:
WorldCat
|
4 |
zzzz
Libraries near you:
WorldCat
|
Book Details
Table of Contents
Preface
Page xi
Preliminaries
Page 1
0.1.
Basics
Page 1
0.2.
Properties of the Integers
0.3.
Z/nZ: The Integers Modulo n
Page 8
Part I.
Group Theory
Page 13
1.
Introduction to Groups
Page 16
1.1.
Basic Axioms and Examples
Page 16
1.2.
Dihedral Groups
Page 23
1.3.
Symmetric Groups
Page 29
1.4.
Matrix Groups
Page 34
1.5.
The Quaternion Group
Page 36
1.6.
Homomorphisms and Isomorphisms
Page 36
1.7.
Group Actions
Page 41
2.
Subgroups
Page 46
2.1.
Definitions and Examples
Page 46
2.2.
Centralizers and Normalizers, Stabilizers and Kernels
Page 49
2.3.
Cyclic Groups and Cyclic Subgroups
Page 54
2.4.
Subgroups Generated by Subsets of a Group
Page 61
2.5.
The Lattice of Subgroups of a Group
Page 66
3.
Quotient Groups and Homomorphisms
Page 73
3.1.
Definitions and Examples
Page 73
3.2.
More on Cosets and Lagrange's Theorem
Page 89
3.3.
The Isomorphism Theorems
Page 97
3.4.
Composition Series and the Holder Program
Page 101
3.5.
Transpositions and the Alternating Group
Page 106
4.
Group Actions
Page 112
4.1.
Group Actions and Permutation Representations
Page 112
4.2.
Groups Acting on Themselves by Left Multiplication-Cayley's Theorem
Page 118
4.3.
Groups Acting on Themselves by Conjugation-The Class Equation
Page 122
4.4.
Automorphism
Page 133
4.5.
The Sylow Theorems
Page 139
4.6.
The Simplicity of An
Page 149
5.
Direct and Semidirect Products and Abelian Groups
Page 152
5.1.
Direct Products
Page 142
5.2.
The Fundamental Theorem of Finitely Generated Ableian Groups
Page 158
5.3.
Table of Groups of Small Order
Page 167
5.4.
Recognizing Direct Products
Page 169
5.5.
Semidirect Products
Page 175
6.
Further Topics in Group Theory
Page 188
6.1.
p-groups, Nilpotent Groups, and Solvable Groups
Page 188
6.2.
Applications in Groups of Medium Order
Page 201
6.3.
A Word on Free Groups
Page 215
Part II.
Ring Theory
Page 222
7.
Introduction to Rings
Page 223
7.1.
Basic Definitions and Exmaples
Page 223
7.2.
Examples: Polynomial Rings, Matri Rings, and Group Rings
Page 233
7.3.
Ring Homomorphisms and Quotinet Rings
Page 239
7.4.
Properties of Ideals
Page 251
7.5.
Rings of fractions
Page 260
7.6.
The Chinese Remainder Theorem
Page 265
8.
Euclidean Domains, Principla Ideal Domains and Unique Factorization Domains
Page 270
8.1.
Euclidean Domains
Page 270
8.2.
Principal Ideal Domains (P.I.D.s)
Page 279
8.3.
Unique Factorization Domains (U.F.D.s)
Page 283
9.
Polynomial Rings
Page 295
9.1.
Definitions and Basic Properties
Page 295
9.2.
Polynomial Rings over Fields I
Page 299
9.3.
Polynomial Rings that are Unique Factorization Domains
Page 303
9.4.
Irreducibility Criteria
Page 307
9.5.
Polynomial rings over Fields II
Page 313
9.6.
Polynomials in Several Variables over a Field and Grobner Bases
Page 315
Part III.
Modules and Vector Spaces
Page 336
10.
Introduction to Module Theory
Page 337
10.1.
Basic Definitions and Examples
Page 337
10.2.
Quotient Modules and Module Homomorphisms
Page 345
10.3.
Generation of Modules, Direct Sums, and Free Modules
Page 351
10.4.
Tensor Product of Modules
Page 359
10.5.
Exact Sequences - Projective, Injective, and Flat Modules
Page 378
11.
vector Spaces
Page 408
11.1.
Definitions and Basic Theory
Page 408
11.2.
The Matrix of a Linear Transformation
Page 415
11.3.
Dual Vector Spaces
Page 431
11.4.
Determinants
Page 435
11.5.
Tensor Algebras, Symmetric and Exterior Algebras
Page 441
12.
Modules over Principal Ideal Domains
Page 456
12.1.
The Basic Theory
Page 458
12.2.
The Rational Canonical From
Page 472
12.3.
The Jordan Canonical From
Page 491
13.
Field Theory
Page 510
13.1.
Basic Theory of field Extensions
Page 510
13.2.
Algebraic extensions
Page 520
13.3.
Classical Straightedge and Compass Constructions
Page 531
13.4.
Splitting Fields and Algebraic Closures
Page 536
13.5.
Separable and Inseparable Extensions
Page 545
13.6.
Cyclotomic Polynomials and Extensions
Page 552
14.
Galios Theory
Page 558
14.1.
Basic Definitions
Page 558
14.2.
The Fundamental Theorem of Galios Theory
Page 567
14.3.
Finite Fields
Page 585
14.4.
Composite Extensions and Simple Extensions
Page 591
14.5.
Cyclotomic Extensions and Albelian Extensions over Q
Page 596
14.6.
Galois Groups of Polynomials
Page 606
14.7.
Solvable and Radical Extensions: Insolvability of the Quintic
Page 625
14.8.
Computation of the Galois Groups over Q
Page 640
14.9.
Transcendental Extensions, Inseparable Extensions, Infinite Galois Groups
Page 645
Part V [sic].
An Introduction to Comutative Rings, Algebraic Geometry, and Homological Algebra
Page 655
15.
Commutative Rings and Algebraic Geometry
Page 656
15.1.
Noetherian Rings and Affine Algebraic Sets
Page 656
15.2.
Radicals and Affine Varieties
Page 673
15.3.
Integral Extensions and Hilbert's Nullstellensatz
Page 691
15.4.
Localization
Page 706
15.5.
The Prime Spectrum of a Ring
Page 731
16.
Artinian Rings, Discrete Valuation Rings, and Dedekind Domains
Page 750
16.1.
Artinian Rings
Page 750
16.2.
Discrete Valuation Rings
Page 755
16.3.
Dedekind Domains
Page 764
17.
Introduction to Homological Algebra and Group Cohomology
Page 776
17.1.
Introduction to Homological Algebra - Ext and Tor
Page 777
17.2.
The Cohomology of Groups
Page 798
17.3.
Crossed Homomorphisms and H1(G, A)
Page 814
17.4.
Group Extensions, Factor Sets and H2(G, A)
Page 824
Part VI.
Introduction to the Representation Theory of Finite Groups
Page 839
18.
Representation Theory and Character Theory
Page 840
18.1.
Linear Actions and Modules over Group Rings
Page 840
18.2.
Wedderburn's Theorem and Some Consequences
Page 854
18.3.
Character Theory and the Orthogonality Relations
Page 864
19.
Examples and Applications of Character Theory
Page 880
19.1.
Characters of Groups of Small Order
Page 880
19.2.
Theorems of Burnside and Hall
Page 886
19.3.
Introduction to the Theory of Induced Characters
Page 892
Appendix I.
Cartesian Products and Zorn's Lemma
Page 905
Appendix II.
Category Theory
Page 911
Index
Page 919
Edition Notes
Includes index.
Classifications
The Physical Object
ID Numbers
Community Reviews (0)
Feedback?December 29, 2022 | Edited by Tom Morris | merge authors |
December 29, 2022 | Edited by MARC Bot | import existing book |
August 1, 2020 | Edited by ImportBot | import existing book |
December 4, 2010 | Edited by Open Library Bot | Added subjects from MARC records. |
December 9, 2009 | Created by WorkBot | add works page |