A survey of modern algebra:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York
Macmillan
1965
London Collier-Macmillan |
| Ausgabe: | 3. ed. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | X, 437 S. graph. Darst. |
Internformat
MARC
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| 245 | 1 | 0 | |a A survey of modern algebra |c Garrett Birkhoff ; Saunders Mac Lane |
| 250 | |a 3. ed. | ||
| 264 | 1 | |a New York |b Macmillan |c 1965 | |
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Datensatz im Suchindex
| _version_ | 1804116358936395776 |
|---|---|
| adam_text | Contents
Preface
I. The Integers i
1. Commutative Rings; Integral Domains 1
2. Elementary Properties of Domains 3
3. Properties of Order 7
4. Well ordering Principle 10
5. Finite Induction; Laws of Exponents 11
6. Divisibility 14
7. The Euclidean Algorithm 15
8. Fundamental Theorem of Arithmetic 20
9. Congruences 22
10. The Rings Zn 25
1 1. Some Basic Concepts of Logic 28
12. Isomorphisms and Automorphisms 30
II. Rational Numbers and Fields 33
1. Definition of a Field 33
2. Construction of the Rationals 36
3. Simultaneous Linear Equations 40
4. Ordered Fields 45
5. Postulates for the Positive Integers 47
6. Peano Postulates 50
III. Polynomials 53
1. Polynomial Forms 53
2. Polynomial Functions 56
3. Zero divisors and Commutative Rings 60
4. Polynomials in Several Variables 62
5. The Division Algorithm 64
6. Units and Associates 66
7. Irreducible Polynomials 68
8. Unique Factorization Theorem 70
9. Other Domains with Unique Factorization 74
10. Eisenstein s Irreducibility Criterion 77
1 1. Partial Fractions 79
IV. Real Numbers 83
1. Dilemma of Pythagoras 83
2. Upper and Lower Bounds 84
3. Postulates for Real Numbers 86
vii
viii Contents
4. Roots of Polynomial Equations 89
5. Dedekind Cuts 91
V. Complex Numbers 95
1. Definition 95
2. The Complex Plane 98
3. Fundamental Theorem of Algebra 101
4. Conjugate Numbers and Real Polynomials 103
5. Quadratic and Cubic Equations 105
6. Solution of Quartic by Radicals 107
7. Equations of Stable Type 108
VI. Groups no
1. Symmetries of the Square 110
2. Groups of Transformations 112
3. Further Examples 116
4. Abstract Groups 118
5. Isomorphism 122
6. Cyclic Groups 125
7. Subgroups 127
8. Lagrange s Theorem 130
9. Permutation Groups 133
10. Even and Odd Permutations 136
1 1. Homomorphisrns 138
1 2. Automorphisms; Conjugate Elements 140
13. Quotient Groups 143
14. Equivalence and Congruence Relations 145
VII. Vectors and Vector Spaces 149
1. Vectors in a Plane 149
2. Generalizations 150
3. Vector Spaces and Subspaces 152
4. Linear Independence and Dimension 155
5. Matrices and Row equivalence 159
6. Tests for Linear Dependence 162
7. Vector Equations; Homogeneous Equations 166
8. Bases and Coordinate Systems 170
9. Inner Products 175
10. Euclidean Vector Spaces 177
1 1. Normal Orthogonal Bases 179
12. Linear Functions and Dual Spaces 182
1 3. Bilinear Functions and Tensor Products 187
VIII. The Algebra of Matrices 189
1. Linear Transformations and Matrices 189
2. Matrix Addition 194
Contents ix
3. Matrix Multiplication 196
4. Diagonal, Permutation, and Triangular Matrices 200
5. Rectangular Matrices 203
6. Inverses 207
7. Rank and Nullity 212
8. Elementary Matrices 215
9. Equivalence and Canonical Form 219
10. Quaternions 222
11. Linear Algebras 225
IX. Linear Groups 229
1. Change of Basis 229
2. Similar Matrices and Characteristic Vectors 237
3. The Full Linear and Affine Groups 236
4. The Orthogonal and Euclidean Groups 240
5. Invariants and Canonical Forms 244
6. Linear and Bilinear Forms 247
7. Quadratic Forms 249
8. Quadratic Forms under the Full Linear Group 252
9. Real Quadratic Forms under the Full Linear Group 254
10. Quadratic Forms under the Orthogonal Group 257
11. Quadrics under the Affine and Euclidean Groups 267
1 2. Unitary and Hermitian Matrices 264
13. Affine Geometry 268
14. Projective Geometry 275
X. Determinants and Canonical Forms 280
1. Definition and Elementary Properties of Determinants 280
2. Products of Determinants 285
3. Determinants as Volumes 288
4. The Characteristic Polynomial 297
5. The Minimal Polynomial 296
6. Cayley Hamilton Theorem 299
7. Invariant Subspaces and Reducibility 301
8. First Decomposition Theorem 304
9. Second Decomposition Theorem 307
10. Rational and Jordan Canonical Forms 309
XI. Boolean Algebras and Lattices 313
1. Basic Definition 313
2. Laws: Analogy with Arithmetic 314
3. Boolean Algebra 317
4. Deduction of Other Basic Laws 319
5. Canonical Forms of Boolean Polynomials 322
6. Partial Orderings 325
7. Lattices 327
8. Representation by Sets 330
x Contents
XII. Transfinite Arithmetic 334
1. Numbers and Sets 334
2. Countable Sets 336
3. Other Cardinal Numbers 336
4. Addition and Multiplication of Cardinals 342
5. Exponentiation 343
XIII. Rings and Ideals 346
1. Rings 346
2. Homomorphisms 347
3. Quotient Rings 357
4. Algebra of Ideals 354
5. Polynomial Ideals 356
6. Ideals in Linear Algebras 360
7. The Characteristic of a Ring 367
8. Characteristics of Fields 364
XIV. Algebraic Number Fields 366
1. Algebraic and Transcendental Extensions 366
2. Elements Algebraic over a Field 368
3. The Adjunction of Roots 377
4. Degrees and Finite Extensions 375
5. Iterated Algebraic Extensions 377
6. Algebraic Numbers 38 7
7. Gaussian Integers 384
8. Algebraic Integers 387
9. Sums and Products of Integers 389
10. Factorization of Quadratic Integers 392
XV. Galois Theory 395
1. Root Fields for Equations 395
2. The Galois Group 397
3. Separable and Inseparable Polynomials 402
4. Properties of the Galois Group 404
5. Subgroups and Subfields 408
6. Finite Fields 477
7. Irreducible Cubic Equations 474
8. Insolvability of Quintic Equations 478
Bibliography 423
List of Special Symbols 426
Index 429
|
| any_adam_object | 1 |
| author | Birkhoff, Garrett 1911-1996 Mac Lane, Saunders 1909-2005 |
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| author_sort | Birkhoff, Garrett 1911-1996 |
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| building | Verbundindex |
| bvnumber | BV001920594 |
| classification_rvk | QH 140 SK 200 |
| ctrlnum | (OCoLC)144582399 (DE-599)BVBBV001920594 |
| dewey-full | 512.02 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.02 |
| dewey-search | 512.02 |
| dewey-sort | 3512.02 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Wirtschaftswissenschaften |
| edition | 3. ed. |
| format | Book |
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| indexdate | 2024-07-09T15:37:15Z |
| institution | BVB |
| language | English |
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| spelling | Birkhoff, Garrett 1911-1996 Verfasser (DE-588)11882564X aut A survey of modern algebra Garrett Birkhoff ; Saunders Mac Lane 3. ed. New York Macmillan 1965 London Collier-Macmillan X, 437 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Algebra (DE-588)4001156-2 gnd rswk-swf 1\p (DE-588)4151278-9 Einführung gnd-content Algebra (DE-588)4001156-2 s DE-604 Mac Lane, Saunders 1909-2005 Verfasser (DE-588)118575953 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001251723&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Birkhoff, Garrett 1911-1996 Mac Lane, Saunders 1909-2005 A survey of modern algebra Algebra (DE-588)4001156-2 gnd |
| subject_GND | (DE-588)4001156-2 (DE-588)4151278-9 |
| title | A survey of modern algebra |
| title_auth | A survey of modern algebra |
| title_exact_search | A survey of modern algebra |
| title_full | A survey of modern algebra Garrett Birkhoff ; Saunders Mac Lane |
| title_fullStr | A survey of modern algebra Garrett Birkhoff ; Saunders Mac Lane |
| title_full_unstemmed | A survey of modern algebra Garrett Birkhoff ; Saunders Mac Lane |
| title_short | A survey of modern algebra |
| title_sort | a survey of modern algebra |
| topic | Algebra (DE-588)4001156-2 gnd |
| topic_facet | Algebra Einführung |
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