Abstract algebra / David S. Dummit, Richard M. Foote.

Dummit, David S. (Author). Foote, Richard M. (Added Author).

Record details

ISBN: 0471433349
Physical Description: xii, 932 p. : il. ; 25 cm.
Edition: 3rd ed.
Publisher: Hoboken, N. J. : John Wiley & Sons, c2004.

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General Note:	Incluye índice.
Language Note:	English

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Algebra abstracta.

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Location	Call Number / Copy Notes	Barcode	Shelving Location	Status	Due Date
Biblioteca Ipicyt	QA162 D8 A2 2004	APL00309	Coleccion General	Available	-

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Biblioteca Ipicyt

QA162 D8 A2 2004

APL00309

Coleccion General

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Table of Contents

	Preface	xi
	Preliminaries	1
0.1.	Basics	1
0.2.	Properties of the Integers
0.3.	Z/nZ: The Integers Modulo n	8
Part I.	Group Theory	13
1.	Introduction to Groups	16
1.1.	Basic Axioms and Examples	16
1.2.	Dihedral Groups	23
1.3.	Symmetric Groups	29
1.4.	Matrix Groups	34
1.5.	The Quaternion Group	36
1.6.	Homomorphisms and Isomorphisms	36
1.7.	Group Actions	41
2.	Subgroups	46
2.1.	Definitions and Examples	46
2.2.	Centralizers and Normalizers, Stabilizers and Kernels	49
2.3.	Cyclic Groups and Cyclic Subgroups	54
2.4.	Subgroups Generated by Subsets of a Group	61
2.5.	The Lattice of Subgroups of a Group	66
3.	Quotient Groups and Homomorphisms	73
3.1.	Definitions and Examples	73
3.2.	More on Cosets and Lagrange's Theorem	89
3.3.	The Isomorphism Theorems	97
3.4.	Composition Series and the Holder Program	101
3.5.	Transpositions and the Alternating Group	106
4.	Group Actions	112
4.1.	Group Actions and Permutation Representations	112
4.2.	Groups Acting on Themselves by Left Multiplication-Cayley's Theorem	118
4.3.	Groups Acting on Themselves by Conjugation-The Class Equation	122
4.4.	Automorphism	133
4.5.	The Sylow Theorems	139
4.6.	The Simplicity of An	149
5.	Direct and Semidirect Products and Abelian Groups	152
5.1.	Direct Products	142
5.2.	The Fundamental Theorem of Finitely Generated Ableian Groups	158
5.3.	Table of Groups of Small Order	167
5.4.	Recognizing Direct Products	169
5.5.	Semidirect Products	175
6.	Further Topics in Group Theory	188
6.1.	p-groups, Nilpotent Groups, and Solvable Groups	188
6.2.	Applications in Groups of Medium Order	201
6.3.	A Word on Free Groups	215
Part II.	Ring Theory	222
7.	Introduction to Rings	223
7.1.	Basic Definitions and Exmaples	223
7.2.	Examples: Polynomial Rings, Matri Rings, and Group Rings	233
7.3.	Ring Homomorphisms and Quotinet Rings	239
7.4.	Properties of Ideals	251
7.5.	Rings of fractions	260
7.6.	The Chinese Remainder Theorem	265
8.	Euclidean Domains, Principla Ideal Domains and Unique Factorization Domains	270
8.1.	Euclidean Domains	270
8.2.	Principal Ideal Domains (P.I.D.s)	279
8.3.	Unique Factorization Domains (U.F.D.s)	283
9.	Polynomial Rings	295
9.1.	Definitions and Basic Properties	295
9.2.	Polynomial Rings over Fields I	299
9.3.	Polynomial Rings that are Unique Factorization Domains	303
9.4.	Irreducibility Criteria	307
9.5.	Polynomial rings over Fields II	313
9.6.	Polynomials in Several Variables over a Field and Grobner Bases	315
Part III.	Modules and Vector Spaces	336
10.	Introduction to Module Theory	337
10.1.	Basic Definitions and Examples	337
10.2.	Quotient Modules and Module Homomorphisms	345
10.3.	Generation of Modules, Direct Sums, and Free Modules	351
10.4.	Tensor Product of Modules	359
10.5.	Exact Sequences - Projective, Injective, and Flat Modules	378
11.	vector Spaces	408
11.1.	Definitions and Basic Theory	408
11.2.	The Matrix of a Linear Transformation	415
11.3.	Dual Vector Spaces	431
11.4.	Determinants	435
11.5.	Tensor Algebras, Symmetric and Exterior Algebras	441
12.	Modules over Principal Ideal Domains	456
12.1.	The Basic Theory	458
12.2.	The Rational Canonical From	472
12.3.	The Jordan Canonical From	491
13.	Field Theory	510
13.1.	Basic Theory of field Extensions	510
13.2.	Algebraic extensions	520
13.3.	Classical Straightedge and Compass Constructions	531
13.4.	Splitting Fields and Algebraic Closures	536
13.5.	Separable and Inseparable Extensions	545
13.6.	Cyclotomic Polynomials and Extensions	552
14.	Galios Theory	558
14.1.	Basic Definitions	558
14.2.	The Fundamental Theorem of Galios Theory	567
14.3.	Finite Fields	585
14.4.	Composite Extensions and Simple Extensions	591
14.5.	Cyclotomic Extensions and Albelian Extensions over Q	596
14.6.	Galois Groups of Polynomials	606
14.7.	Solvable and Radical Extensions: Insolvability of the Quintic	625
14.8.	Computation of the Galois Groups over Q	640
14.9.	Transcendental Extensions, Inseparable Extensions, Infinite Galois Groups	645
Part V [sic].	An Introduction to Comutative Rings, Algebraic Geometry, and Homological Algebra	655
15.	Commutative Rings and Algebraic Geometry	656
15.1.	Noetherian Rings and Affine Algebraic Sets	656
15.2.	Radicals and Affine Varieties	673
15.3.	Integral Extensions and Hilbert's Nullstellensatz	691
15.4.	Localization	706
15.5.	The Prime Spectrum of a Ring	731
16.	Artinian Rings, Discrete Valuation Rings, and Dedekind Domains	750
16.1.	Artinian Rings	750
16.2.	Discrete Valuation Rings	755
16.3.	Dedekind Domains	764
17.	Introduction to Homological Algebra and Group Cohomology	776
17.1.	Introduction to Homological Algebra - Ext and Tor	777
17.2.	The Cohomology of Groups	798
17.3.	Crossed Homomorphisms and H1(G, A)	814
17.4.	Group Extensions, Factor Sets and H2(G, A)	824
Part VI.	Introduction to the Representation Theory of Finite Groups	839
18.	Representation Theory and Character Theory	840
18.1.	Linear Actions and Modules over Group Rings	840
18.2.	Wedderburn's Theorem and Some Consequences	854
18.3.	Character Theory and the Orthogonality Relations	864
19.	Examples and Applications of Character Theory	880
19.1.	Characters of Groups of Small Order	880
19.2.	Theorems of Burnside and Hall	886
19.3.	Introduction to the Theory of Induced Characters	892
Appendix I.	Cartesian Products and Zorn's Lemma	905
Appendix II.	Category Theory	911
	Index	919

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