Introduction to abstract algebra:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
CRC Press
2009
|
| Series: | Textbooks in mathematics
A Chapman & Hall book |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | X, 327 S. graph. Darst. |
| ISBN: | 9781420063714 |
Staff View
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| 245 | 1 | 0 | |a Introduction to abstract algebra |c Jonathan D. H. Smith |
| 264 | 1 | |a Boca Raton [u.a.] |b CRC Press |c 2009 | |
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Record in the Search Index
| _version_ | 1804137987997433856 |
|---|---|
| adam_text | Contents
NUMBERS
1
1.1
Ordering numbers
....................... 1
1.2
The Well-Ordering Principle
................. 3
1.3
Divisibility
........................... 5
1.4
The Division Algorithm
.................... 6
1.5
Greatest common divisors
................... 9
1.6
The Euclidean Algorithm
................... 10
1.7
Primes and irreducibles
.................... 13
1.8
The Fundamental Theorem of Arithmetic
.......... 14
1.9
Exercises
............................ 17
1.10
Study projects
......................... 22
1.11
Notes
.............................. 23
FUNCTIONS
25
2.1
Specifying functions
...................... 25
2.2
Composite functions
...................... 27
2.3
Linear functions
........................ 28
2.4
Semigroups of functions
.................... 29
2.5
Injectivity and surjectivity
.................. 31
2.6
Isomorphisms
.......................... 34
2.7
Groups of permutations
.................... 36
2.8
Exercises
............................ 39
2.9
Study projects
......................... 43
2.10
Notes
.............................. 46
2.11
Summary
............................ 47
EQUIVALENCE
49
3.1
Kernel and equivalence relations
............... 49
3.2
Equivalence classes
....................... 51
3.3
Rational numbers
....................... 53
3.4
The First Isomorphism Theorem for Sets
.......... 56
3.5
Modular arithmetic
...................... 58
3.6
Exercises
............................ 61
3.7
Study projects
......................... 63
3.8
Notes
.............................. 66
VI
4
GROUPS AND MONOIDS
67
4.1
Semigroups
........................... 67
4.2
Monoids
............................. 69
4.3
Groups
............................. 71
4.4
Componentwise structure
................... 73
4.5
Powers
............................. 77
4.6
Submonoids and subgroups
.................. 78
4.7
Cosets
.............................. 82
4.8
Multiplication tables
...................... 84
4.9
Exercises
............................ 87
4.10
Study projects
......................... 91
4.11
Notes
.............................. 94
5
HOMOMORPHISMS
95
5.1
Homomorphisms
........................ 95
5.2
Normal subgroups
....................... 98
5.3
Quotients
............................ 101
5.4
The First Isomorphism Theorem for Groups
........ 104
5.5
The Law of Exponents
.................... 106
5.6
Cayley s Theorem
....................... 109
5.7
Exercises
............................ 112
5.8
Study projects
......................... 116
5.9
Notes
.............................. 125
6
RINGS
127
6.1
Rings
.............................. 127
6.2
Distributivity
.......................... 131
6.3
Subrings
............................ 133
6.4
Ring homomorphisms
..................... 135
6.5
Ideals
.............................. 137
6.6
Quotient rings
......................... 139
6.7
Polynomial rings
........................ 140
6.8
Substitution
.......................... 145
6.9
Exercises
............................ 147
6.10
Study projects
......................... 151
6.11
Notes
.............................. 156
7
FIELDS
157
7.1
Integral domains
........................ 157
7.2
Degrees
............................. 160
7.3
Fields
.............................. 162
7.4
Polynomials over fields
.................... 164
7.5
Principal ideal domains
.................... 167
7.6
Irreducible polynomials
.................... 170
7.7 Lagrange
interpolation
.................... 173
vu
7.8
Fields of fractions
....................... 175
7.9
Exercises
............................ 178
7.10
Study projects
......................... 182
7.11
Notes
.............................. 184
8
FACTORIZATION
185
8.1
Factorization in integral domains
............... 185
8.2
Noetherian domains
...................... 188
8.3
Unique factorization domains
................. 190
8.4
Roots of polynomials
..................... 193
8.5
Splitting fields
......................... 196
8.6
Uniqueness of splitting fields
................. 198
8.7
Structure of finite fields
.................... 202
8.8
Galois fields
.......................... 204
8.9
Exercises
............................ 206
8.10
Study projects
......................... 210
8.11
Notes
.............................. 213
9
MODULES
215
9.1
Endomorphisms
........................ 215
9.2
Representing a ring
...................... 219
9.3
Modules
............................. 220
9.4
Submodules
........................... 223
9.5
Direct sums
........................... 227
9.6
Free modules
.......................... 231
9.7
Vector spaces
.......................... 235
9.8
Abelian groups
......................... 240
9.9
Exercises
............................ 243
9.10
Study projects
......................... 248
9.11
Notes
.............................. 251
10
GROUP ACTIONS
253
10.1
Actions
............................. 253
10.2
Orbits
.............................. 256
10.3
Transitive actions
....................... 258
10.4
Fixed points
.......................... 262
10.5
Faithful actions
......................... 265
10.6
Cores
.............................. 267
10.7
Alternating groups
....................... 270
10.8
Sylow Theorems
........................ 273
10.9
Exercises
............................ 277
10.10
Study projects
......................... 283
10.11
Notes
.............................. 286
Vlil
11
QUASIGROUPS
287
11.1
Quasigroups
.......................... 287
11.2
Latin
squares
.......................... 289
11.3
Division
............................. 293
11.4
Quasigroup homomorphisms
................. 297
11.5
Quasigroup homotopies
.................... 301
11.6
Principal
isotopy
........................ 304
11.7
Loops ..............................
306
11.8
Exercises
............................ 311
11.9
Study projects
......................... 315
11.10
Notes
.............................. 318
Index
319
|
| adam_txt |
Contents
NUMBERS
1
1.1
Ordering numbers
. 1
1.2
The Well-Ordering Principle
. 3
1.3
Divisibility
. 5
1.4
The Division Algorithm
. 6
1.5
Greatest common divisors
. 9
1.6
The Euclidean Algorithm
. 10
1.7
Primes and irreducibles
. 13
1.8
The Fundamental Theorem of Arithmetic
. 14
1.9
Exercises
. 17
1.10
Study projects
. 22
1.11
Notes
. 23
FUNCTIONS
25
2.1
Specifying functions
. 25
2.2
Composite functions
. 27
2.3
Linear functions
. 28
2.4
Semigroups of functions
. 29
2.5
Injectivity and surjectivity
. 31
2.6
Isomorphisms
. 34
2.7
Groups of permutations
. 36
2.8
Exercises
. 39
2.9
Study projects
. 43
2.10
Notes
. 46
2.11
Summary
. 47
EQUIVALENCE
49
3.1
Kernel and equivalence relations
. 49
3.2
Equivalence classes
. 51
3.3
Rational numbers
. 53
3.4
The First Isomorphism Theorem for Sets
. 56
3.5
Modular arithmetic
. 58
3.6
Exercises
. 61
3.7
Study projects
. 63
3.8
Notes
. 66
VI
4
GROUPS AND MONOIDS
67
4.1
Semigroups
. 67
4.2
Monoids
. 69
4.3
Groups
. 71
4.4
Componentwise structure
. 73
4.5
Powers
. 77
4.6
Submonoids and subgroups
. 78
4.7
Cosets
. 82
4.8
Multiplication tables
. 84
4.9
Exercises
. 87
4.10
Study projects
. 91
4.11
Notes
. 94
5
HOMOMORPHISMS
95
5.1
Homomorphisms
. 95
5.2
Normal subgroups
. 98
5.3
Quotients
. 101
5.4
The First Isomorphism Theorem for Groups
. 104
5.5
The Law of Exponents
. 106
5.6
Cayley's Theorem
. 109
5.7
Exercises
. 112
5.8
Study projects
. 116
5.9
Notes
. 125
6
RINGS
127
6.1
Rings
. 127
6.2
Distributivity
. 131
6.3
Subrings
. 133
6.4
Ring homomorphisms
. 135
6.5
Ideals
. 137
6.6
Quotient rings
. 139
6.7
Polynomial rings
. 140
6.8
Substitution
. 145
6.9
Exercises
. 147
6.10
Study projects
. 151
6.11
Notes
. 156
7
FIELDS
157
7.1
Integral domains
. 157
7.2
Degrees
. 160
7.3
Fields
. 162
7.4
Polynomials over fields
. 164
7.5
Principal ideal domains
. 167
7.6
Irreducible polynomials
. 170
7.7 Lagrange
interpolation
. 173
vu
7.8
Fields of fractions
. 175
7.9
Exercises
. 178
7.10
Study projects
. 182
7.11
Notes
. 184
8
FACTORIZATION
185
8.1
Factorization in integral domains
. 185
8.2
Noetherian domains
. 188
8.3
Unique factorization domains
. 190
8.4
Roots of polynomials
. 193
8.5
Splitting fields
. 196
8.6
Uniqueness of splitting fields
. 198
8.7
Structure of finite fields
. 202
8.8
Galois fields
. 204
8.9
Exercises
. 206
8.10
Study projects
. 210
8.11
Notes
. 213
9
MODULES
215
9.1
Endomorphisms
. 215
9.2
Representing a ring
. 219
9.3
Modules
. 220
9.4
Submodules
. 223
9.5
Direct sums
. 227
9.6
Free modules
. 231
9.7
Vector spaces
. 235
9.8
Abelian groups
. 240
9.9
Exercises
. 243
9.10
Study projects
. 248
9.11
Notes
. 251
10
GROUP ACTIONS
253
10.1
Actions
. 253
10.2
Orbits
. 256
10.3
Transitive actions
. 258
10.4
Fixed points
. 262
10.5
Faithful actions
. 265
10.6
Cores
. 267
10.7
Alternating groups
. 270
10.8
Sylow Theorems
. 273
10.9
Exercises
. 277
10.10
Study projects
. 283
10.11
Notes
. 286
Vlil
11
QUASIGROUPS
287
11.1
Quasigroups
. 287
11.2
Latin
squares
. 289
11.3
Division
. 293
11.4
Quasigroup homomorphisms
. 297
11.5
Quasigroup homotopies
. 301
11.6
Principal
isotopy
. 304
11.7
Loops .
306
11.8
Exercises
. 311
11.9
Study projects
. 315
11.10
Notes
. 318
Index
319 |
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| index_date | 2024-07-02T21:55:58Z |
| indexdate | 2024-07-09T21:21:02Z |
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| isbn | 9781420063714 |
| language | English |
| lccn | 2008027689 |
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| spelling | Smith, Jonathan D. H. Verfasser aut Introduction to abstract algebra Jonathan D. H. Smith Boca Raton [u.a.] CRC Press 2009 X, 327 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Textbooks in mathematics A Chapman & Hall book 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 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016717984&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Smith, Jonathan D. H. Introduction to abstract algebra Algebra, Abstract Universelle Algebra (DE-588)4061777-4 gnd |
| 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_exact_search_txtP | Introduction to abstract algebra |
| title_full | Introduction to abstract algebra Jonathan D. H. Smith |
| title_fullStr | Introduction to abstract algebra Jonathan D. H. Smith |
| title_full_unstemmed | Introduction to abstract algebra Jonathan D. H. Smith |
| 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 |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016717984&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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