A graduate course in algebra: Volume 2
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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: | xv, 400 Seiten Diagramme |
| ISBN: | 9789813142664 9789813142671 |
Staff View
MARC
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| 245 | 1 | 0 | |a A graduate course in algebra |n Volume 2 |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] | |
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| 500 | |a Enthält Literaturverzeichnis und Index | ||
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface and Acknowledgments
xi
7 Multilinear Algebra 1
7.1 Multilinear Functions ................................. 1
7.2 The Symmetric Algebra . . . ........................... 2
7.3 The Exterior Algebra................................... 6
7.3.1 Determinants................................... 13
7.3.2 Cramer’s Rule ............................... 16
7.3.3 Orientation.................................... 18
7.3.4 The Volume Form on a real vector space...... 24
7.3.5 The Hodge star (*) Operator.................... 26
7.4 Liouville’s Formula................................... 35
7.5 Grassmannians......................................... 37
7.5.1 Exercises...................................... 43
8 Symplectic Geometry 45
8.1 Symplectic Vector Spaces............................... 45
8.1.1 Symplectic Subspaces............................ 51
8.2 Linear Complex Structures and Sp(n, R) ............... 58
8.3 The Topology of the Symplectic Groups.................. 62
8.4 Transvections.......................................... 65
8.4.1 Eigenvalues of a Symplectic Matrix............ 65
8.4.2 Sp(n, R) is generated by Transvections........ 68
8.5 J. Williamson’s Normal Form............................ 71
8.6 Wirtinger’s Inequality................................. 74
v
vi Contents
8.7 M. Gromov’s Non-squeezing Theorem........................ 76
8.7.1 Exercises ....................................... 81
9 Commutative Rings with Identity 87
9.1 Principal Ideal Domains................................. 87
9.1.1 A Principal Ideal Domain which is not Euclidean 90
9.2 Unique Factorization Domains ........................... 93
9.2.1 Kaplansky’s Characterization of a UFD............ 95
9.3 The Power Series Ring R[[x]] 97
9.3.1 Exercises....................................... 105
9.4 Prime Ideals.......................................... 105
9.4.1 The Prime Avoidance Lemma....................... 106
9.5 Local Rings............................................ 109
9.6 Localization........................................... 112
9.6.1 Localization at a Prime Ideal................... 120
9.6.2 Localization of Modules......................... 122
9.6.3 Exercises ...................................... 125
9.7 Noetherian Rings....................................... 126
9.7.1 Hilbert’s Basis Theorem......................... 131
9.8 Artinian Rings......................................... 134
9.8.1 The Hopkins-Levitzki Theorem.................... 136
9.9 Noetherian and Artinian Modules........................ 137
9.9.1 An Artinian but not Noetherian Module........... 142
9.10 Krull Dimension........................................ 143
9.11 Nakayama’s Lemma....................................... 147
9.11.1 Applications of Nakayama’s Lemma................ 148
9.11.2 Three Variants of Nakayama’s Lemma.............. 151
9.12 The Radical of an Ideal................................ 154
9.13 The Nilradical and the Jacobson Radical of a Ring . . . 156
9.13.1 The Jacobson Radical............................ 156
9.13.2 The Nilradical ................................. 159
9.14 The Hilbert Nullstellensatz Theorem................... 165
9.15 Algebraic Varieties and the Zariski Topology......... 171
9.15.1 Zariski Density................................. 174
9.16 The Completion of a Ring.............................. 177
Contents vii
9.16.1 Exercises......................................... 182
10 Valuations and the p-adic Numbers 185
10.1 Valuations............................................... 185
10.2 Absolute Value or Norm................................... 187
10.3 Non-Archimedean Absolute Values vs. Valuations .... 193
10.3.1 The Ring of Integers......................... 195
10.4 Absolute Values on Q ............................... 197
10.4.1 Ostrowski’s Theorem............................... 199
10.5 Ultra-metric Spaces ..................................... 201
10.6 The Completion of Q...................................... 206
10.6.1 The Fields Qp are all Non-Isomorphic......... 213
10.7 The Algebraic Definition of Zp ....................... 213
10.8 The p-adic Numbers, Qp................................ 216
10.9 The p-adic Topology...................................... 218
10.9.1 Zp and Qp as Topological Groups and Rings . . . 224
10.10 The Geometry of Qp...................................... 228
10.11 Extensions of Valuations................................ 231
10.12 An Application: Monsky’s Theorem........................ 233
10.12.1 Exercises ....................................... 236
11 Galois Theory 239
11.1 Field Extensions......................................... 239
11.2 Algebraic Extensions and Splitting Fields................ 240
11.3 Finite Groups of Automorphisms of a Field........... 243
11.3.1 Exercises........................................ 249
11.4 Normal Separable Extensions and the Galois Group . . 249
11.5 The Fundamental Theorem of Galois Theory................. 256
11.6 Consequences of Galois Theory............................ 259
11.6.1 Non Solvability of Equations of Degree 5 . . . 259
11.6.2 Classical Ruler and Compass Constructions . . . 262
11.6.3 The Fundamental Theorem of Algebra............... 264
11.7 Solution of the General Cubic and Quartic Equations . . 266
11.8 Some Preliminaries from Algebraic Number Theory . . . 269
11.8.1 Exercises......................................... 272
Vili
Contents
12 Group Representations 275
12.1 Introduction.......................................... 275
12.1.1 The Regular Representation..................... 278
12.1.2 The Character of a Representation.............. 281
12.2 The Schur Orthogonality Relations..................... 281
12.3 Characters and Central Functions...................... 286
12.4 Induced Representations............................... 292
12.4.1 Some Consequences of Frobenius Reciprocity . . 296
12.5 Estimates and Divisibility Properties................. 300
12.6 Burnside’s paqb Theorem............................... 302
12.7 Clifford’s Theorem.................................... 305
12.8 Some Applications of Real Representation Theory . . . 307
12.9 Finitely Generated Linear Groups ..................... 310
12.9.1 Exercises ..................................... 314
12.10 Pythagorean Triples (continued)...................... 314
12.10.1 Exercises .................................... 317
13 Representations of Associative Algebras 321
13.1 The Double Commutant Theorem ......................... 323
13.2 Burnside’s Theorem.................................... 324
13.3 Semi-simple Algebras and Wedderburn’s Theorem . . . 329
13.3.1 The Big Wedderburn Theorem.................... 335
13.4 Division Algebras..................................... 338
13.4.1 Jacobson’s Theorem............................ 344
13.4.2 Frobenius’ Theorem............................ 345
13.4.3 The 4-Square Theorem.......................... 349
13.4.4 The 24-cell Polytope.......................... 352
13.4.5 Exercises .................................... 360
13.4.6 The Symmetry Groups of the Platonic Solids . . 361
Appendix 365
Appendix A: Every Field has an Algebraic Closure 365
Appendix B: The Fundamental Theorem of Algebra 369
Contents ix
Appendix C: Irrational and Transcendental Numbers 371
Bibliography 385
Index 397
|
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| author | Farmakis, Ioannis Moskowitz, Martin A. |
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| institution | BVB |
| isbn | 9789813142664 9789813142671 |
| language | English |
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| physical | xv, 400 Seiten Diagramme |
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| spelling | Farmakis, Ioannis Verfasser (DE-588)1043300783 aut A graduate course in algebra Volume 2 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] xv, 400 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Enthält Literaturverzeichnis und Index Moskowitz, Martin A. Verfasser (DE-588)114556628 aut (DE-604)BV044326889 2 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=029730258&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 2 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 2 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 2 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=029730258&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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