A graduate course in algebra: Volume 1
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New Jersey
World Scientific
[2017]
|
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Enthält Literaturverzeichnis und Index |
| Physical Description: | xvii, 436 Seiten Illustrationen, Diagramme |
| ISBN: | 9789813142626 9789813142633 |
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MARC
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| 245 | 1 | 0 | |a A graduate course in algebra |n Volume 1 |c Ioannis Farmakis (Department of Mathematics, Brooklyn College, City University of New York, USA), Martin Moskowitz (Ph.D. Program in Mathematics, CUNY Graduate Center, City University of New York, USA) |
| 264 | 1 | |a New Jersey |b World Scientific |c [2017] | |
| 264 | 0 | |c [2017] | |
| 264 | 4 | |c © 2017 | |
| 264 | 4 | |c © 2017 | |
| 300 | |a xvii, 436 Seiten |b Illustrationen, Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Enthält Literaturverzeichnis und Index | ||
| 700 | 1 | |a Moskowitz, Martin A. |e Verfasser |0 (DE-588)114556628 |4 aut | |
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Record in the Search Index
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| adam_text | Contents
Preface and Acknowledgments xi
0 Introduction 1
0.1 Sets................................................. 1
0.2 Cartesian Product.................................... 3
0.3 Relations............................................. 3
0.4 Partially Ordered Sets............................... 6
0.4.1 The Principle of Induction..................... 6
0.4.2 Transfinite Induction.......................... 9
0.4.3 Permutations................................... 9
0.5 The set (Z,+, x).................................... 11
0.5.1 The Fundamental Theorem of Arithmetic .... 13
0.5.2 The Euler Formula and Riemann Zeta Function . 17
0.5.3 The Fermat Numbers ............. 21
0.5.4 Pythagorean Triples........................... 23
1 Groups 27
1.1 The Concept of a Group.............................. 27
1.2 Examples of Groups.................................. 31
1.2.1 The Quaternion Group ............. 33
1.2.2 The Dihedral Group......................... . 35
1.3 Subgroups........................................... 38
1.3.0.1 Exercises............................. 40
1.4 Quotient Groups..................................... 41
1.4.1 Cosets........................................ 41
v
vi Contents
1.5 Modular Arithmetic ................................... 44
1.5.1 Chinese Remainder Theorem....................... 48
1.5.2 Fermat’s Little Theorem......................... 50
1.5.3 Wilson’s Theorem................................ 55
1.5.3.1 Exercises............................... 57
1.6 Automorphisms, Characteristic and Normal Subgroups . 58
1.7 The Center of a Group, Commutators...................... 62
1.8 The Three Isomorphism Theorems.......................... 63
1.9 Groups of Low Order..................................... 66
1.9.0.1 Exercises................................ 70
1.10 Direct and Semi-direct Products........................ 71
1.10.1 Direct Products................................. 71
1.10.2 Semi-Direct Products ........................... 75
1.11 Exact Sequences of Groups........................... 79
1.12 Direct and Inverse Limits of Groups................. 82
1.13 Free Groups............................................ 89
1.14 Some Features of Abelian Groups....................... 92
1.14.1 Torsion Groups.................................. 92
1.14.2 Divisible and Injective Groups.................. 96
1.14.3 Priifer Groups ................................ 100
1.14.4 Structure Theorem for Divisible Groups......... 102
1.14.5 Maximal Subgroups.............................. 104
1.14.5.1 Exercises . ........................... 105
2 Further Topics in Group Theory 109
2.1 Composition Series..................................... 109
2.2 Solvability, Nilpotency................................ 112
2.2.1 Solvable Groups................................ 112
2.2.2 Nilpotent Groups............................... 117
2.2.2.1 Exercises.............................. 120
2.3 Group Actions........................................ 121
2.3.1 Stabilizer and Orbit........................... 124
2.3.2 Transitive Actions............................. 126
2.3.3 Some Examples of Transitive Actions ........... 128
2.3.3.1 The Real Sphere S ”՜1.................. 128
Contents vii
2.3.3.2 The Complex Sphere S2n~l............ 129
2.3.3.3 The Quaternionic Sphere S4n~՜1 .... 130
2.3.3.4 The Poincaré Upper Half-Plane .... 131
2.3.3.5 Real Projective Space RJPn.......... 132
2.3.3.6 Complex Projective Space CPn........ 133
2.3.3.7 The Grassmann and Flag Varieties . . . 135
2.4 PSL(n, k), An L· Iwasawa’s Double Transitivity Theorem 138
2.5 Imprimitive Actions..................................... 145
2.6 Cauchy’s Theorem........................................ 147
2.7 The Three Sylow Theorems................................ 148
2.7.0. 1 Exercises........................... 152
3 Vector Spaces 155
3.1 Generation, Basis and Dimension......................... 157
3.1.0. 1 Exercises........................... 162
3.2 The First Isomorphism Theorem....................... . 162
3.2.1 Systems of Linear Equations ..................... 164
3.2.2 Cramer’s Rule ................................... 166
3.3 Second and Third Isomorphism Theorems................... 168
3.4 Linear Transformations and Matrices..................... 170
3.4.1 Eigenvalues, Eigenvectors and Diagonalizability . 173
3.4.1.1 The Fibonacci Sequence.............. 175
3.4.2 Application to Matrix Differential Equations . . 178
3.4.2.1 Exercises........................... 181
3.5 The Dual Space.......................................... 181
3.5.1 Annihilators..................................... 183
3.5.2 Systems of Linear Equations Revisited............ 184
3.5.3 The Adjoint of an Operator ...................... 186
3.6 Direct Sums............................................. 187
3.7 Tensor Products....................................... 189
3.7.1 Tensor Products of Linear Operators.............. 194
3.8 Complexification of a Real Vector Space................. 198
3.8.1 Complexifying with Tensor Products............... 201
3.8.2 Real Forms and Complex Conjugation............... 205
Vill
Contents
4 Inner Product Spaces 207
4.0. 1 Gram-Schmidt Orthogonalization................... 213
4.0.1.1 Legendre Polynomials..................... 214
4.0. 2 Bessel’s Inequality and Parseval’s Equation . . . 216
4.1 Subspaces and their Orthocomplements.................... 218
4.2 The Adjoint Operator.................................... 220
4.3 Unitary and Orthogonal Operators........................ 221
4.3.1 Eigenvalues of Orthogonal and Unitary Operators 223
4.4 Symmetric and Hermitian Operators....................... 223
4.4.1 Skew-Symmetric and Skew-Hermitian Operators 225
4.5 The Cayley Transform.................................... 228
4.6 Normal Operators ....................................... 230
4.6.1 The Spectral Theorem............................. 231
4.7 Some Applications to Lie Groups......................... 238
4.7.1 Positive Definite Operators...................... 238
4.7.1.1 Exercises................................ 242
4.7.2 The Topology on 7i+ and *P+...................... 243
4.7.3 The Polar Decomposition.......................... 245
4.7.4 The Iwasawa Decomposition ....................... 247
4.7.4.1 I det I and Volume in Rn............. 250
4.7.5 The Bruhat Decomposition......................... 252
4.8 Gramians................................................ 259
4.9 Schur’s Theorems and Eigenvalue Estimates............... 263
4.10 The Geometry of the Conics............................ 267
4.10.1 Polarization of Symmetric Bilinear Forms .... 267
4.10.2 Classification of Quadric Surfaces under Aff(V) . 268
5 Rings, Fields and Algebras 277
5.1 Preliminary Notions..................................... 277
5.2 Subrings and Ideals..................................... 284
5.3 Homomorphisms........................................... 285
5.3.1 The Three Isomorphism Theorems................... 286
5.3.2 The Characteristic of a Field.................... 287
5.4 Maximal Ideals.......................................... 289
5.4.1 Prime Ideals .................................... 291
Contents ix
5.5 Euclidean Rings ...................................... 293
5.5.1 Vieta’s Formula and the Discriminant....... 297
5.5.2 The Chinese Remainder Theorem.................. 298
5.6 Unique Factorization.................................. 303
5.6.1 Fermat’s Two-Square Thm. Gaussian Primes . 305
5.7 The Polynomial Ring................................... 309
5.7.1 Gauss’ Lemma and Eisenstein’s Criterion .... 310
5.7.2 Cyclotomic Polynomials..................... 313
5.7.3 The Formal Derivative...................... 316
5.7.4 The Fundamental Thm. of Symmetric
Polynomials.................................... 320
5.7.5 There is No Such Thing as a Pattern........ 324
5.7.6 Non-Negative Polynomials................... 325
5.7.6.1 Exercises.............................. 328
5.8 Finite Fields and Wedderburn’s Little Theorem......... 330
5.8.1 Application to Projective Geometry......... 333
5.9 k- Algebras........................................... 335
5.9.1 Division Algebras.......................... 341
5.9.1.1 Exercises.............................. 344
6 A-Modules 347
6.1 Generalities on R-Modules............................. 347
6.1.1 The Three Isomorphism Theorems for Modules . 350
6.1.2 Direct Products and Direct Sums............ 351
6.2 Homological Algebra................................... 353
6.2.1 Exact Sequences............................ 354
6.2.2 Free Modules............................... 362
6.2.3 The Tensor Product of A-modules............ 366
6.3 A-Mo chiles vs. Vector Spaces......................... 373
6.4 Finitely Generated Modules over a Euclidean Ring . . . 375
6.4.0.1 Exercises.............................. 379
6.5 Applications to Linear Transformations................ 379
6.6 The Jordan Canonical Form and Jordan Decomposition 385
6.6.1 The Minimal Polynomial (continued...)...... 389
6.6.2 Families of Commuting Operators ............... 391
X
Contents
6.6.3 Additive Multiplicative Jordan Decompositions 393
6.7 The Jordan-Holder Theorem for /¿-Modules....... 397
6.8 The Fitting Decomposition and Krull-Schmidt Theorem 400
6.9 Schur’s Lemma and Simple Modules............... 403
Appendix 407
Appendix A: Pell’s Equation 407
Appendix B: The Kronecker Approximation Theorem 414
Appendix C: Some Groups of Automorphisms 416
Bibliography 419
Index
429
|
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| author | Farmakis, Ioannis Moskowitz, Martin A. |
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| spelling | Farmakis, Ioannis Verfasser (DE-588)1043300783 aut A graduate course in algebra Volume 1 Ioannis Farmakis (Department of Mathematics, Brooklyn College, City University of New York, USA), Martin Moskowitz (Ph.D. Program in Mathematics, CUNY Graduate Center, City University of New York, USA) New Jersey World Scientific [2017] [2017] © 2017 xvii, 436 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Enthält Literaturverzeichnis und Index Moskowitz, Martin A. Verfasser (DE-588)114556628 aut (DE-604)BV044326889 1 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029730252&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Farmakis, Ioannis Moskowitz, Martin A. A graduate course in algebra |
| title | A graduate course in algebra |
| title_auth | A graduate course in algebra |
| title_exact_search | A graduate course in algebra |
| title_full | A graduate course in algebra Volume 1 Ioannis Farmakis (Department of Mathematics, Brooklyn College, City University of New York, USA), Martin Moskowitz (Ph.D. Program in Mathematics, CUNY Graduate Center, City University of New York, USA) |
| title_fullStr | A graduate course in algebra Volume 1 Ioannis Farmakis (Department of Mathematics, Brooklyn College, City University of New York, USA), Martin Moskowitz (Ph.D. Program in Mathematics, CUNY Graduate Center, City University of New York, USA) |
| title_full_unstemmed | A graduate course in algebra Volume 1 Ioannis Farmakis (Department of Mathematics, Brooklyn College, City University of New York, USA), Martin Moskowitz (Ph.D. Program in Mathematics, CUNY Graduate Center, City University of New York, USA) |
| title_short | A graduate course in algebra |
| title_sort | a graduate course in algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029730252&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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