Fundamental structures of algebra:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York
McGraw-Hill
1963
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 585 S. |
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MARC
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Record in the Search Index
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| adam_text | Contents
Preface v
List of Special Symbol* »
1. Binary Operations and Groups 1
1. INTRODUCTION 1
2. SETS AND MAPPINGS 1
3. BINARY OPERATIONS 5
4. THE ASSOCIATIVE AXIOM 8
5. THE COMMUTATIVE AXIOM 11
G. GROUPS 12
7. ISOMORPHISMS AND HOMOMORPHISMS IS
8. RESTATEMENT OF THE GROUP AXIOMS 20
9. SYSTEMS WITH TWO BINARY OPERATIONS: RINGS, INTEGRAL DOMAINS,
FIELDS 21
2. Rings, Integral Domains, the Integers 28
1. INTRODUCTION 28
2. SYSTEMS WITH TWO BINARY OPERATIONS: RINGS AND INTEGRAL DOMAINS 28
3. ORDERED INTEGRAL DOMAINS 32
4. THE SYSTEM OF INTEGERS 38
5. SOME COMMENTS 39
6. FINITE AND COUNTABLE SETS 42
7. MATHEMATICAL INDUCTION AND SOME OF ITS APPLICATIONS 43
8. SOME ELEMENTARY NUMBER THEORY 54
9. NOTATION FOR INTEGERS 62
10. MORE ELEMENTARY NUMBER THEORY: CONGRUENCES 66
11. PROOF OF THEOREM 4.4 77
x Contents
3. Fields, the Rational Numbers 81
1. INTRODUCTION 81
2. FIELDS 81
3. THE FIELD OF RATIONAL NUMBERS 85
4. DECIMALS 89
5. THE BINOMIAL THEOREM 95
k. The Real number System 100
1. INTRODUCTION 100
2. CAUCHY SEQUENCES AND LIMITS 102
3. THE FIELD OF REAL NUMBERS 108
4. SOME PROPERTIES OF R 111
5. The Field of Complex Numbers 118
1. THE SQUARE ROOT OF 1 118
2. A CONSTRUCTION OF C; QUATERNIONS 122
3. A GEOMETRIC INTERPRETATION OF ADDITION AND MULTIPLICATION OF
COMPLEX NUMBERS 125
4. CAUCHY SEQUENCES AND INFINITE SERIES IN C 129
6. Polynomials 133
1. INTRODUCTION 133
2. INDETERMINATES, OR VARIABLES 133
3. FACTORIZATION OF POLYNOMIALS 140
4. ROOTS OF POLYNOMIALS 148
5. POLYNOMIALS IN SEVERAL VARIABLES 153
6. POLYNOMIALS OF DEGREE LESS THAN 5 158
7. Rational Functions 163
1. INTRODUCTION 163
2. RATIONAL FUNCTIONS 163
3. PARTIAL FRACTIONS 165
Contents xi
8. Vector Spaces and A fine Spaces 171
1. INTRODUCTION 171
2. THE BASIC DEFINITIONS 171
3. SOME CONSEQUENCES OF THE AXIOMS 172
4. SOME IMPORTANT EXAMPLES 174
5. SUBSPACES 176
6. LINEAR INDEPENDENCE AND DIMENSION 178
7. A THEOREM ON LINEAR EQUATIONS ISO
8. ON THE DIMENSION OF VECTOR SPACES 182
9. BASE VECTORS 183
10. AFFINE SPACES 187
11. EUCLIDEAN SPACES 198
12. ANALYTIC GEOMETRY 205
9. Linear Transformations and Matrices 216
1. INTRODUCTION 216
2. A NOTATIONAL CONVENTION 216
3. LINEAR MAPPINGS 217
4. OPERATIONS ON LINEAR MAPPINGS 221
5. LINEAR TRANSFORMATIONS AND MATRICES 225
6. OPERATIONS ON MATRICES 231
7. CHANGE OF BASE 238
8. RANK OF A MATRIX; LINEAR EQUATIONS; SUBSPACES 242
9. REDUCTION TO DIAGONAL FORM 246
10. QUOTIENT SPACES 258
11. MODULES 261
10. Groups and Permutations 266
1. INTRODUCTION 266
2. BASIC PROPERTIES 266
3. PERMUTATIONS 268
4. SUBGROUPS AND QUOTIENT GROUPS 275
xii Contents
5. TRANSFORMATION GROUPS; SYLOW S THEOREMS 282
6. THE JORDAN HOLDER THEOREM 291
7. FINITE ABELIAN GROUPS 298
11. Determinants 304
1. INTRODUCTION 304
2. AXIOMS FOR DETERMINANTS 305
3. SOME APPLICATIONS 311
4. THE CHARACTERISTIC POLYNOMIAL 317
5. EIGENVALUES AND EIGENVECTORS 324
6. DETERMINANTS AS VOLUMES 333
12. Rings of Operators and Differential Equations 341
1. INTRODUCTION 341
2. RINGS AND HOMOMORPHISMS 341
3. HOMOMORPHISMS OF RINGS 344
4. THE DIFFERENTIATION OPERATOR 347
5. SOME DIFFERENTIATION FORMULAS 3S2
6. LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS 354
7. FINDING PARTICULAR AND GENERAL SOLUTION 361
8. TRIGONOMETRIC FUNCTIONS 367
9. SYSTEMS OF EQUATIONS 370
10. ONE PARAMETER GROUPS AND INFINITESIMAL GENERATORS 377
13. The Jordan Normal Form 379
1. INTRODUCTION 379
2. ELEMENTARY LINEAR MAPPINGS 380
3. DIRECT SUM DECOMPOSITIONS 382
4. NILPOTENT MAPPINGS 389
5. CHARACTERISTIC SUBSPACES 395
6. THE JORDAN NORMAL FORM 399
7. UNIQUENESS OF THE JORDAN NORMAL FORM 406
Contents xiii
8. THE PROBLEM OF SIMILARITY 408
9. ELEMENTARY DIVISORS 411
10. ELEMENTARY DIVISORS AND SIMILARITY 420
11. MODULES, TORSION ORDERS, AND THE RATIONAL CANONICAL FORM 423
12. FINITELY GENERATED ABELIAN GROUPS 431
1U Quadratic and Hermitian Forms 433
1. INTRODUCTION 433
2. LINEAR FUNCTIONS; DUAL SPACES 433
3. BILINEAR FUNCTIONS 438
4. QUADRATIC FORMS 441
5. REDUCTION TO DIAGONAL FORM 446
S. HERMITIAN FORMS; UNITARY MAPPINGS 452
7. EUCLIDEAN VECTOR SPACES 459
8. ORTHONORMAL BASES 462
9. FOURIER SERIES, BESSEL S INEQUALITY 467
10. THE EIGENVALUES OF A HERMITIAN MATRIX 470
11. SIMULTANEOUS DIAGONALIZATION OF TWO HERMITIAN FORMS 474
12. UNITARY MATRICES 478
13. VECTOR PRODUCTS IN ORIENTED 3 SPACE 479
14. ANALYTIC GEOMETRY IN n DIMENSIONS 486
15. Quotient Structures 494
1. MAPPINGS 494
2. RELATIONS 496
3. QUOTIENT SET 497
4. BINARY OPERATIONS ON QUOTIENT SETS 499
5. THE CONSTRUCTION OF THE FIELD OF QUOTIENTS OF AN INTEGRAL DOMAIN 503
6. THE CONSTRUCTION OF THE FIELD OF REAL NUMBERS FROM THE FIELD OF
RATIONAL NUMBERS 504
7. THE CONSTRUCTION OF A FIELD CONTAINING A ROOT OF A POLYNOMIAL 508
8. A PARADOX TO AVOID 510
9. BERNSTEIN S THEOREM ON CARDINAL NUMBERS 510
xiv Contents
16. Tensors 514
1. INTRODUCTION 514
2. TENSOR PRODUCTS 514
3. TENSOR PRODUCTS OF MORE THAN TWO FACTORS 519
4. TENSOR PRODUCTS OF MAPPINGS 524
5. THE TENSOR ALGEBRA OF A VECTOR SPACE 527
6. BASES AND COMPONENTS 534
7. CONTRACTION OF TENSORS 538
8. SYMMETRY PROPERTIES 543
9. THE METRIC 547
10. THE EXTERIOR ALGEBRA 551
11. PLUCKER COORDINATES; DUALITY 561
12. SKEW SYMMETRIC TENSORS 572
Index 575
|
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| spelling | Mostow, George D. Verfasser aut Fundamental structures of algebra George D. Mostow ; Joseph Sampson and Jean-Pierre Meyer New York McGraw-Hill 1963 XVI, 585 S. txt rdacontent n rdamedia nc rdacarrier Algèbre Algèbre ram Algebra Algebras, Linear Gruppentheorie (DE-588)4072157-7 gnd rswk-swf Grundlage (DE-588)4158388-7 gnd rswk-swf Algebra (DE-588)4001156-2 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Algebra (DE-588)4001156-2 s DE-604 Grundlage (DE-588)4158388-7 s Gruppentheorie (DE-588)4072157-7 s Sampson, Joseph H. Verfasser aut Meyer, Jean-Pierre Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002589566&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mostow, George D. Sampson, Joseph H. Meyer, Jean-Pierre Fundamental structures of algebra Algèbre Algèbre ram Algebra Algebras, Linear Gruppentheorie (DE-588)4072157-7 gnd Grundlage (DE-588)4158388-7 gnd Algebra (DE-588)4001156-2 gnd |
| subject_GND | (DE-588)4072157-7 (DE-588)4158388-7 (DE-588)4001156-2 (DE-588)4151278-9 |
| title | Fundamental structures of algebra |
| title_auth | Fundamental structures of algebra |
| title_exact_search | Fundamental structures of algebra |
| title_full | Fundamental structures of algebra George D. Mostow ; Joseph Sampson and Jean-Pierre Meyer |
| title_fullStr | Fundamental structures of algebra George D. Mostow ; Joseph Sampson and Jean-Pierre Meyer |
| title_full_unstemmed | Fundamental structures of algebra George D. Mostow ; Joseph Sampson and Jean-Pierre Meyer |
| title_short | Fundamental structures of algebra |
| title_sort | fundamental structures of algebra |
| topic | Algèbre Algèbre ram Algebra Algebras, Linear Gruppentheorie (DE-588)4072157-7 gnd Grundlage (DE-588)4158388-7 gnd Algebra (DE-588)4001156-2 gnd |
| topic_facet | Algèbre Algebra Algebras, Linear Gruppentheorie Grundlage Einführung |
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