A course on abstract algebra:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Singapore [u.a.]
World Scientific
2010
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 359 S. graph. Darst. |
| ISBN: | 9789814271882 9814271888 |
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MARC
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| 245 | 1 | 0 | |a A course on abstract algebra |c Minking Eie ; Shou-Te Chang |
| 264 | 1 | |a Singapore [u.a.] |b World Scientific |c 2010 | |
| 300 | |a XII, 359 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-018734395 | ||
Record in the Search Index
| _version_ | 1804140854067068928 |
|---|---|
| adam_text | Contents
Preface
v
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
3.4
Generating Sets
.........................
48
ix
x
Contents
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
................. 87
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
....................... 133
9-3
Structure Theorem of Finite Abelian Groups
........ 137
Review Exercises
........................... 142
Contents xi
і
10 Sylow
Theorems and Applications
143
10.1
Group Actions
......................... 144
10.2
Sylow Theorems
........................ 151
Review Exercises
........................... 157
11
Introduction to Group Presentations
159
11.1
Free Groups and Free Abelian Groups
............ 160
11.2
Generators and Relations
................... 165
11.3
Classification of Finite Groups of Small Orders
....... 170
Review Exercises
........................... 175
12
Types of Rings
177
12.1
Definitions and Examples
................... 178
12.2
Matrix Rings
.......................... 185
Review Exercises
........................... 191
13
Ideals and Quotient Rings
193
13.1
Ideals
.............................. 194
13.2
Quotient Rings
......................... 198
Review Exercises
........................... 203
14
Ring Homomorphisms
205
14.1
Ring Homomorphisms
..................... 206
14.2
Direct Products of Rings
.................... 211
14.3
The Quotient Field of an Integral Domain
.......... 216
Review Exercises
........................... 222
15
Polynomial Rings
223
15.1
Polynomial Rings in the Indeterminates
........... 224
15.2
Properties of the Polynomial Rings of One Variable
..... 228
15.3
Principal Ideal Domains and Euclidean Domains
...... 233
Review Exercises
........................... 237
16
Factorization
239
16.1
Irreducible and Prime Elements
................ 240
16.2
Unique Factorization Domains
................ 245
16.3
Polynomial Extensions of Factorial Domains
......... 253
xii
Contents
Review Exercises
........................... 259
17
Vector Spaces Over an Arbitrary Field
261
17.1
A Brief Review on Vector Spaces
............... 262
17.2
A Brief Review on Linear Transformations
.......... 266
Review Exercises
........................... 272
18
Field Extensions
273
18.1
Algebraic or Transcendental?
................. 274
18.2
Finite and Algebraic Extensions
................ 278
18.3
Construction with Straightedge and Compass
........ 284
Review Exercises
........................... 294
19
All About Roots
295
19.1
Zeros of Polynomials
...................... 296
19.2
Uniqueness of Splitting Fields
................. 299
19.3
Algebraically Closed Fields
.................. 303
19.4
Multiplicity of Roots
...................... 305
19.5
Finite Fields
.......................... 309
Review Exercises
........................... 314
20
Galois Pairing
315
20.1
Galois Groups
.......................... 316
20.2
The Fixed Subfields of a Galois Group
............ 321
20.3
Fundamental Theorem of Galois Pairing
........... 326
Review Exercises
........................... 331
21
Applications of the Galois Pairing
333
21.1
Fields
ofinvariants
....................... 334
21.2
Solvable Groups
........................ 338
21.3
Insolvability of the Quintic
.................. 345
Review Exercises
........................... 350
Index
351
|
| any_adam_object | 1 |
| author | Eie, Minking 1952- Chang, Shou-Te 19XX- |
| author_GND | (DE-588)140942785 (DE-588)141170972 |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
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| dewey-search | 512 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV035876726 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:06:35Z |
| institution | BVB |
| isbn | 9789814271882 9814271888 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-018734395 |
| oclc_num | 565491440 |
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| owner | DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-824 |
| owner_facet | DE-355 DE-BY-UBR DE-19 DE-BY-UBM DE-824 |
| physical | XII, 359 S. graph. Darst. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| 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 Singapore [u.a.] World Scientific 2010 XII, 359 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Abstract algebra 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 Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018734395&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Eie, Minking 1952- Chang, Shou-Te 19XX- A course on abstract algebra Abstract algebra 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 |
| title_fullStr | A course on abstract algebra Minking Eie ; Shou-Te Chang |
| title_full_unstemmed | A course on abstract algebra Minking Eie ; Shou-Te Chang |
| title_short | A course on abstract algebra |
| title_sort | a course on abstract algebra |
| topic | Abstract algebra Algebra (DE-588)4001156-2 gnd |
| topic_facet | Abstract algebra Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018734395&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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