Introduction to abstract algebra:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Hoboken, NJ
Wiley
2012
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| Edition: | 4. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXIV, 535 S. graph. Darst. |
| ISBN: | 9781118135358 |
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Record in the Search Index
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| adam_text | Titel: Introduction to abstract algebra
Autor: Nicholson, W. Keith
Jahr: 2012
Contents
PREFACE ix
ACKNOWLEDGMENTS xvii
NOTATION USED IN THE TEXT xix
A SKETCH OF THE HISTORY OF ALGEBRA TO 1929 xxiii
0 Preliminaries 1
0.1 Proofs / 1
0.2 Sets / 5
0.3 Mappings / 9
0.4 Equivalences / 17
1 Integers and Permutations 23
1.1 Induction / 24
1.2 Divisors and Prime Factorization / 32
1.3 Integers Modulo n / 42
1.4 Permutations / 53
1.5 An Application to Cryptography / 67
2 Groups 69
2.1 Binary Operations / 70
2.2 Groups / 76
2.3 Subgroups / 86
2.4 Cyclic Groups and the Order of an Element / 90
vi Contents
2.5 Homomorphisms and Isomorphisms / 99
2.6 Cosets and Lagrange s Theorem / 108
2.7 Groups of Motions and Symmetries / 117
2.8 Normal Subgroups / 122
2.9 Factor Groups / 131
2.10 The Isomorphism Theorem / 137
2.11 An Application to Binary Linear Codes / 143
3 Rings 159
3.1 Examples and Basic Properties / 160
3.2 Integral Domains and Fields / 171
3.3 Ideals and Factor Rings / 180
3.4 Homomorphisms / 189
3.5 Ordered Integral Domains / 199
4 Polynomials 202
4.1 Polynomials / 203
4.2 Factorization of Polynomials Over a Field / 214
4.3 Factor Rings of Polynomials Over a Field / 227
4.4 Partial Fractions / 236
4.5 Symmetrie Polynomials / 239
4.6 Formal Construction of Polynomials / 248
5 Factorization in Integral Domains 251
5.1 Irreducibles and Unique Factorization / 252
5.2 Principal Ideal Domains / 264
6 Fields 274
6.1 Vector Spaces / 275
6.2 Algebraic Extensions / 283
6.3 Splitting Fields / 291
6.4 Finite Fields / 298
6.5 Geometrie Constructions / 304
6.6 The Fundamental Theorem of Algebra / 308
6.7 An Application to Cyclic and BCH Codes / 310
7 Modules over Principal Ideal Domains 324
7.1 Modules / 324
7.2 Modules Over a PID / 335
Contents vii
8 p-Groups and the Sylow Theorems 349
8.1 Products and Factors / 350
8.2 Cauchy s Theorem / 357
8.3 Group Actions / 364
8.4 The Sylow Theorems / 371
8.5 Semidirect Products / 379
8.6 An Application to Combinatorics / 382
9 Series of Subgroups 388
9.1 The Jordan-Hölder Theorem / 389
9.2 Solvable Groups / 395
9.3 Nilpotent Groups / 401
10 Galois Theory 412
10.1 Galois Groups and Separability / 413
10.2 The Main Theorem of Galois Theory / 422
10.3 Insolvability of Polynomials / 434
10.4 Cyclotomic Polynomials and Wedderburn s Theorem / 442
11 Finiteness Conditions for Rings and Modules 447
11.1 Wedderburn s Theorem / 448
11.2 The Wedderburn-Artin Theorem / 457
Appendices 471
Appendix A Complex Numbers / 471
Appendix B Matrix Algebra / 478
Appendix C Zorn s Lemma / 486
Appendix D Proof of the Recursion Theorem / 490
BIBLIOGRAPHY 492
SELECTED ANSWERS 495
INDEX 523
|
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| isbn | 9781118135358 |
| language | English |
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| physical | XXIV, 535 S. graph. Darst. |
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| spelling | Nicholson, W. Keith 1938- Verfasser (DE-588)1022416685 aut Introduction to abstract algebra W. Keith Nicholson 4. ed. Hoboken, NJ Wiley 2012 XXIV, 535 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Algebra, Abstract Universelle Algebra (DE-588)4061777-4 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Universelle Algebra (DE-588)4061777-4 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025112050&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
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| subject_GND | (DE-588)4061777-4 (DE-588)4151278-9 |
| title | Introduction to abstract algebra |
| title_auth | Introduction to abstract algebra |
| title_exact_search | Introduction to abstract algebra |
| title_full | Introduction to abstract algebra W. Keith Nicholson |
| title_fullStr | Introduction to abstract algebra W. Keith Nicholson |
| title_full_unstemmed | Introduction to abstract algebra W. Keith Nicholson |
| title_short | Introduction to abstract algebra |
| title_sort | introduction to abstract algebra |
| topic | Algebra, Abstract Universelle Algebra (DE-588)4061777-4 gnd |
| topic_facet | Algebra, Abstract Universelle Algebra Einführung |
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