A course on abstract algebra:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New Jersey
World Scientific
[2018]
|
| Edition: | Second edition |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes index |
| Physical Description: | xiii, 417 pages Illustrationen 24 cm |
| ISBN: | 9789813229624 9813229624 |
Staff View
MARC
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| 245 | 1 | 0 | |a A course on abstract algebra |c Minking Eie, Shou-Te Chang, National Chung Cheng University, Taiwan |
| 250 | |a Second edition | ||
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| 264 | 4 | |c © 2018 | |
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Record in the Search Index
| _version_ | 1804178149600133120 |
|---|---|
| adam_text | Contents
Preface v
About the Authors viii
1 Preliminaries 1
1.1 Basic Ideas of Set Theory................................. 2
1.2 Functions................................................. 7
1.3 Equivalence Relations and Partitions..................... 11
1.4 A Note on Natural Numbers................................ 14
Review Exercises.............................................. 16
2 Algebraic Structure of Numbers 17
2.1 The Set of Integers...................................... 18
2.2 Congruences of Integers.................................. 21
2.3 Rational Numbers......................................... 28
Review Exercises.............................................. 33
3 Basic Notions of Groups 35
3.1 Definitions and Examples................................. 36
3.2 Basic Properties ........................................ 41
3.3 Subgroups................................................ 45
ix
x Contents
3.4 Generating Sets........................................ 48
Review Exercises............................................ 51
4 Cyclic Groups 53
4.1 Cyclic Groups.......................................... 54
4.2 Subgroups of Cyclic Groups............................. 57
Review Exercises............................................ 63
5 Permutation Groups 65
5.1 Symmetric Groups....................................... 66
5.2 Dihedral Groups........................................ 71
5.3 Alternating Groups..................................... 76
Review Exercises............................................ 79
6 Counting Theorems 81
6.1 Lagrange’s Theorem..................................... 82
6.2 Conjugacy Classes of a Group........................... 86
Review Exercises............................................ 93
7 Group Homomorphisms 95
7.1 Examples and Basic Properties.......................... 96
7.2 Isomorphisms........................................... 99
7.3 Cayley’s Theorem ..................................... 105
Review Exercises........................................... 108
8 The Quotient Group 109
8.1 Normal Subgroups...................................... 110
8.2 Quotient Groups....................................... 114
8.3 Fundamental Theorem of Group Homomorphisms ...... 119
Review Exercises........................................... 125
9 Finite Abelian Groups 127
9.1 Direct Products of Groups............................. 128
9.2 Cauchy’s Theorem...................................... 135
9.3 Structure Theorem of Finite Abelian Groups............ 139
Review Exercises........................................... 143
Contents xi
10 Group Actions 145
10.1 Definition and Basic Properties......................... 146
10.2 Orbits and Stabilizers.................................. 151
10.3 Burnside’s Formula...................................... 156
Review Exercises............................................. 161
11 Sylow Theorems and Applications 163
11.1 The Three Sylow Theorems.............................. 164
11.2 Applications of Sylow Theorems ......................... 169
Review Exercises............................................. 173
12 Introduction to Group Presentations 175
12.1 Free Groups and Free Abelian Groups..................... 176
12.2 Generators and Relations................................ 181
12.3 Classification of Finite Groups of Small Orders....... 185
Review Exercises............................................. 192
13 Types of Rings 193
13.1 Definitions and Examples................................ 194
13.2 Matrix Rings............................................ 202
Review Exercises............................................. 207
14 Ideals and Quotient Rings 209
14.1 Ideals.................................................. 210
14.2 Quotient Rings.......................................... 214
Review Exercises............................................. 219
15 Ring Homomorphisms 221
15.1 Ring Homomorphisms...................................... 222
15.2 Direct Products of Rings................................ 227
15.3 The Quotient Field of an Integral Domain................ 232
Review Exercises........................................... 238
16 Polynomial Rings 239
16.1 Polynomial Rings in the Indeterminates.................. 240
16.2 Properties of the Polynomial Rings of One Variable...... 245
16.3 Principal Ideal Domains and Euclidean Domains........... 250
Contents
• ·
Xll
Review Exercises ............................................ 253
17 Factorization 255
17.1 Irreducible and Prime Elements.......................... 256
17.2 Unique Factorization Domains ........................... 261
17.3 Polynomial Extensions of Factorial Domains . . . ........269
Review Exercises............................................. 275
18 Introduction to Modules 277
18.1 Modules and Submodules...................................278
18.2 Linear Maps and Quotient Modules........................ 286
18.3 Direct Sums of Modules...................................293
Review Exercises............................................. 298
19 Free Modules 299
19.1 Free Modules............................................ 300
19.2 Determinant ............................................ 307
Review Exercises............................................. 316
20 Vector Spaces over Arbitrary Fields 317
20.1 A Brief Review on Vector Spaces......................... 318
20.2 A Brief Review on Linear Transformations................ 323
Review Exercises..............................................328
21 Field Extensions 329
21.1 Algebraic or Transcendental?............................ 330
21.2 Finite and Algebraic Extensions......................... 334
21.3 Construction with Straightedge and Compass.............. 340
Review Exercises............................................. 350
22 All About Roots 351
22.1 Zeros of Polynomials.....................................352
22.2 Uniqueness of Splitting Fields.......................... 355
22.3 Algebraically Closed Fields............................. 359
22.4 Multiplicity of Roots................................... 361
22.5 Finite Fields .......................................... 366
Review Exercises............................................. 370
Contents
Xlll
23 Galois Pairing 371
23.1 Galois Groups............................................ 372
23.2 The Fixed Subfields of a Galois Group.................... 377
23.3 Fundamental Theorem of Galois Pairing.................... 382
Review Exercises.............................................. 387
24 Applications of the Galois Pairing 389
24.1 Fields of Invariants..................................... 390
24.2 Solvable Groups ......................................... 394
24.3 Insolvability of the Quintic ............................ 401
Review Exercises.............................................. 406
Index 407
|
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| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.02 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | Second edition |
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| id | DE-604.BV044688238 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T07:59:23Z |
| institution | BVB |
| isbn | 9789813229624 9813229624 |
| language | English |
| lccn | 017038817 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-030085204 |
| oclc_num | 1019992411 |
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| owner | DE-739 DE-19 DE-BY-UBM DE-91G DE-BY-TUM |
| owner_facet | DE-739 DE-19 DE-BY-UBM DE-91G DE-BY-TUM |
| physical | xiii, 417 pages Illustrationen 24 cm |
| publishDate | 2018 |
| publishDateSearch | 2018 |
| publishDateSort | 2018 |
| publisher | World Scientific |
| record_format | marc |
| spelling | Eie, Minking 1952- Verfasser (DE-588)140942785 aut A course on abstract algebra Minking Eie, Shou-Te Chang, National Chung Cheng University, Taiwan Second edition New Jersey World Scientific [2018] © 2018 xiii, 417 pages Illustrationen 24 cm txt rdacontent n rdamedia nc rdacarrier Includes index Algebra, Abstract Algebra, Abstract Textbooks Algebra (DE-588)4001156-2 gnd rswk-swf Algebra (DE-588)4001156-2 s DE-604 Chang, Shou-Te 19XX- Verfasser (DE-588)141170972 aut 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=030085204&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Eie, Minking 1952- Chang, Shou-Te 19XX- A course on abstract algebra Algebra, Abstract Algebra, Abstract Textbooks Algebra (DE-588)4001156-2 gnd |
| subject_GND | (DE-588)4001156-2 |
| title | A course on abstract algebra |
| title_auth | A course on abstract algebra |
| title_exact_search | A course on abstract algebra |
| title_full | A course on abstract algebra Minking Eie, Shou-Te Chang, National Chung Cheng University, Taiwan |
| title_fullStr | A course on abstract algebra Minking Eie, Shou-Te Chang, National Chung Cheng University, Taiwan |
| title_full_unstemmed | A course on abstract algebra Minking Eie, Shou-Te Chang, National Chung Cheng University, Taiwan |
| title_short | A course on abstract algebra |
| title_sort | a course on abstract algebra |
| topic | Algebra, Abstract Algebra, Abstract Textbooks Algebra (DE-588)4001156-2 gnd |
| topic_facet | Algebra, Abstract Algebra, Abstract Textbooks Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030085204&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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