Learning abstract algebra with ISETL:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | Undetermined |
| Veröffentlicht: |
New York u.a.
Springer
1994
|
| Ausgabe: | [Ausg. mit] DOS-Diskette |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIX, 252 S. 1 Diskette 3,5" |
| ISBN: | 3540941045 0387941045 |
Internformat
MARC
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| 100 | 1 | |a Dubinsky, Ed |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Learning abstract algebra with ISETL |c Ed Dubinsky ; Uri Leron |
| 250 | |a [Ausg. mit] DOS-Diskette | ||
| 264 | 1 | |a New York u.a. |b Springer |c 1994 | |
| 300 | |a XIX, 252 S. |e 1 Diskette 3,5" | ||
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Datensatz im Suchindex
| _version_ | 1804123526637027328 |
|---|---|
| adam_text | Contents
Comments for the Student xi
Comments for the Instructor xvii
Acknowledgments xxi
1 Mathematical Constructions in ISETL 1
1.1 Using ISETL 1
1.1.1 Activities 1
1.1.2 Getting started 5
1.1.3 Simple objects and operations on them 6
1.1.4 Control statements 7
1.1.5 Exercises 8
1.2 Compound objects and operations on them 11
1.2.1 Activities 11
1.2.2 Tuples 14
1.2.3 Sets 15
1.2.4 Set and tuple formers 16
1.2.5 Set operations 17
1.2.6 Permutations 17
1.2.7 Quantification 18
1.2.8 Miscellaneous ISETL features 20
1.2.9 VISETL 20
1.2.10 Exercises 21
vi Contents
1.3 Functions in ISETL 25
1.3.1 Activities 25
1.3.2 Funcs 31
1.3.3 Alternative syntax for funcs 32
1.3.4 Using funcs to represent situations 33
1.3.5 Funcs for binary operations 33
1.3.6 Funcs to test properties 33
1.3.7 Smaps 34
1.3.8 Procs 35
1.3.9 Exercises 35
2 Groups 39
2.1 Getting acquainted with groups 39
2.1.1 Activities 39
2.1.2 Definition of a group 42
2.1.3 Examples of groups 43
Number systems 43
Integers mod n 45
Symmetric groups 47
Symmetries of the square 49
Groups of matrices 52
2.1.4 Elementary properties of groups 52
2.1.5 Exercises 55
2.2 The modular groups and the symmetric groups ....... 57
2.2.1 Activities 57
2.2.2 The modular groups Zn 60
2.2.3 The symmetric groups Sn 65
Orbits and cycles 68
2.2.4 Exercises 69
2.3 Properties of groups 71
2.3.1 Activities 71
2.3.2 The specific and the general 72
2.3.3 The cancellation law—An illustration of the
abstract method 74
2.3.4 How many groups are there? 75
Classifying groups of order 4 77
2.3.5 Looking ahead—subgroups 79
2.3.6 Summary of examples and non examples of groups . 80
2.3.7 Exercises 81
3 Subgroups 83
3.1 Definitions and examples 83
3.1.1 Activities 83
3.1.2 Subsets of a group 86
Definition of a subgroup 86
Contents vii
3.1.3 Examples of subgroups 88
Embedding one group in another 88
Conjugates 89
Cycle decomposition and conjugates in Sn ¦ . . 91
3.1.4 Exercises 92
3.2 Cyclic groups and their subgroups 94
3.2.1 Activities 94
3.2.2 The subgroup generated by a single element 96
3.2.3 Cyclic groups 100
The idea of the proof 101
3.2.4 Generators 103
Generators of Sn 103
Parity—even and odd permutations 104
Determining the parity of a permutation 105
3.2.5 Exercises 105
3.3 Lagrange s theorem 108
3.3.1 Activities 108
3.3.2 What Lagrange s theorem is all about Ill
3.3.3 Cosets 112
3.3.4 The proof of Lagrange s theorem 113
3.3.5 Exercises 116
4 The Fundamental Homomorphism Theorem 119
4.1 Quotient groups 119
4.1.1 Activities 119
4.1.2 Normal subgroups 121
Multiplying cosets by representatives 124
4.1.3 The quotient group 125
4.1.4 Exercises 126
4.2 Homomorphisms 129
4.2.1 Activities 129
4.2.2 Homomorphisms and kernels 133
4.2.3 Examples 133
4.2.4 Invariants 135
4.2.5 Homomorphisms and normal subgroups 136
An interesting example 137
4.2.6 Isomorphisms 138
4.2.7 Identifications 139
4.2.8 Exercises 141
4.3 The homomorphism theorem 143
4.3.1 Activities 143
4.3.2 The canonical homomorphism 145
4.3.3 The fundamental homomorphism theorem 147
4.3.4 Exercises 150
viii Contents
5 Rings 153
5.1 Rings 153
5.1.1 Activities 153
5.1.2 Definition of a ring 156
5.1.3 Examples of rings 156
5.1.4 Rings with additional properties 157
Integral domains 157
Fields 158
5.1.5 Constructing new rings from old—matrices 159
5.1.6 Constructing new rings from old—polynomials . . . 161
5.1.7 Constructing new rings from old—functions 164
5.1.8 Elementary properties—arithmetic 165
5.1.9 Exercises 165
5.2 Ideals 168
5.2.1 Activities 168
5.2.2 Analogies between groups and rings 170
5.2.3 Subrings 171
Definition of subring 171
5.2.4 Examples of subrings 171
Subrings of Zn and Z 171
Subrings of M(R) 172
Subrings of polynomial rings 172
Subrings of rings of functions 173
5.2.5 Ideals and quotient rings 173
Definition of ideal 173
Examples of ideals 175
5.2.6 Elementary properties of ideals 175
5.2.7 Elementary properties of quotient rings 176
Quotient rings that are integral domains—
prime ideals 176
Quotient rings that are fields—maximal ideals . 177
5.2.8 Exercises 178
5.3 Homomorphisms and isomorphisms 181
5.3.1 Activities 181
5.3.2 Definition of homomorphism and isomorphism . . . 182
Group homomorphisms vs. ring homomorphisms 183
5.3.3 Examples of homomorphisms and isomorphisms . . . 183
Homomorphisms from Zn to Zk 183
Homomorphisms of Z 184
Homomorphisms of polynomial rings 184
Embeddings—Z, Zn as universal subobjects . . 184
The characteristic of an integral domain and
afield 185
5.3.4 Properties of homorphisms 186
Preservation 186
Contents ix
Ideals and kernels of ring homomorphisms ... 186
5.3.5 The fundamental homomorphism theorem 187
The canonical homomorphism 187
The fundamental theorem 187
Homomorphic images of Z, Zn 188
Identification of quotient rings 188
5.3.6 Exercises 190
6 Factorization in Integral Domains 193
6.1 Divisibility properties of integers and polynomials 193
6.1.1 Activities 193
6.1.2 The integral domains 2, Q[x] 198
Arithmetic and factoring 198
The meaning of unique factorization 199
6.1.3 Arithmetic of polynomials 200
Long division of polynomials 200
6.1.4 Division with remainder 202
6.1.5 Greatest Common Divisors and the Euclidean
algorithm 204
6.1.6 Exercises 208
6.2 Euclidean domains and unique factorization 209
6.2.1 Activities 209
6.2.2 Gaussian integers 212
6.2.3 Can unique factorization fail? 214
6.2.4 Elementary properties of integral domains 214
6.2.5 Euclidean domains 218
Examples of Euclidean domains 219
6.2.6 Unique factorization in Euclidean domains 221
6.2.7 Exercises 225
6.3 The ring of polynomials over a field 226
6.3.1 Unique factorization in F[x] 227
6.3.2 Roots of polynomials 228
6.3.3 The evaluation homomorphism 230
6.3.4 Reducible and irreducible polynomials 231
Examples 231
6.3.5 Extension fields 235
Construction of the complex numbers 237
6.3.6 Splitting fields 237
6.3.7 Exercises 239
Index 241
|
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| author_facet | Dubinsky, Ed |
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| id | DE-604.BV009110280 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T17:31:10Z |
| institution | BVB |
| isbn | 3540941045 0387941045 |
| language | Undetermined |
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| owner_facet | DE-824 DE-83 |
| physical | XIX, 252 S. 1 Diskette 3,5" |
| publishDate | 1994 |
| publishDateSearch | 1994 |
| publishDateSort | 1994 |
| publisher | Springer |
| record_format | marc |
| spelling | Dubinsky, Ed Verfasser aut Learning abstract algebra with ISETL Ed Dubinsky ; Uri Leron [Ausg. mit] DOS-Diskette New York u.a. Springer 1994 XIX, 252 S. 1 Diskette 3,5" txt rdacontent n rdamedia nc rdacarrier ISETL (DE-588)4205825-9 gnd rswk-swf Universelle Algebra (DE-588)4061777-4 gnd rswk-swf MS-DOS (DE-588)4114641-4 gnd rswk-swf Universelle Algebra (DE-588)4061777-4 s ISETL (DE-588)4205825-9 s MS-DOS (DE-588)4114641-4 s DE-604 Leron, Uri Sonstige oth HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006040551&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Dubinsky, Ed Learning abstract algebra with ISETL ISETL (DE-588)4205825-9 gnd Universelle Algebra (DE-588)4061777-4 gnd MS-DOS (DE-588)4114641-4 gnd |
| subject_GND | (DE-588)4205825-9 (DE-588)4061777-4 (DE-588)4114641-4 |
| title | Learning abstract algebra with ISETL |
| title_auth | Learning abstract algebra with ISETL |
| title_exact_search | Learning abstract algebra with ISETL |
| title_full | Learning abstract algebra with ISETL Ed Dubinsky ; Uri Leron |
| title_fullStr | Learning abstract algebra with ISETL Ed Dubinsky ; Uri Leron |
| title_full_unstemmed | Learning abstract algebra with ISETL Ed Dubinsky ; Uri Leron |
| title_short | Learning abstract algebra with ISETL |
| title_sort | learning abstract algebra with isetl |
| topic | ISETL (DE-588)4205825-9 gnd Universelle Algebra (DE-588)4061777-4 gnd MS-DOS (DE-588)4114641-4 gnd |
| topic_facet | ISETL Universelle Algebra MS-DOS |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006040551&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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