Modern higher algebra:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Chicago [u.a.]
Univ. of Chicago Press
1965
|
| Ausgabe: | 10. impr. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XII, 319 S. |
Internformat
MARC
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| 100 | 1 | |a Albert, A. Adrian |d 1905-1972 |e Verfasser |0 (DE-588)122621336 |4 aut | |
| 245 | 1 | 0 | |a Modern higher algebra |c by A. Adrian Albert |
| 250 | |a 10. impr. | ||
| 264 | 1 | |a Chicago [u.a.] |b Univ. of Chicago Press |c 1965 | |
| 300 | |a XII, 319 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 0 | 7 | |a Algebra |0 (DE-588)4001156-2 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Algebra |0 (DE-588)4001156-2 |D s |
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Datensatz im Suchindex
| _version_ | 1804135885618282496 |
|---|---|
| adam_text | TABLE OF CONTENTS
CHAPTER PAGE
I. Groups and Rings 1
1. Introduction 1
2. Sets 1
3. Correspondences 2
4. Integers 5
5. Groups 7
6. Equivalence, subgroups 10
7. Transformation groups 12
8. Rings 13
9. Some properties of rings 15
10. Linear sets over a ring 16
11. Sequences 16
12. Polynomials over S 17
II. Rings with a Unity Element 21
1. The ring g 21
2. The characteristic of a ring 22
3. The ring ?l[x] 24
4. Integral domains and fields 26
5. Equivalence, subsystems 29
6. Divisibility 30
7. The integral domain %[x] 32
8. Unique factorization domains 34
9. The domain 3[x] 36
10. The results of the text 39
11. Linear sets over g 40
12. Forms 43
13. Algebras over g 43
III. Matrices 46
1. Rectangular matrices 46
2. Multiplication of matrices 48
3. Determinants 52
4. The rank of matrices 53
5. The algebra 5KB . .* 55
6. Elementary transformations over 3 59
7. Equivalence of matrices in a field 8 62
8. Linear equations 67
9. Bilinear forms 70
10. Equivalence in g[X] 70
ix
x TABLE OF CONTENTS
OOAPTSB Ti.O*
IV. SIMILARITY OF SQUARE MATRICES 75
1. linear transformations and similar matrices 75
2. The meaning of canonical forms 76
3. Scalar polynomials 78
4. Canonical forms 80
5. Elementary divisors 82
6. Nilpotent matrices 87
7. Idempotent matrices 88
8. The automorphisms of 2K. 89
9. Another basis of 2R» 91
V. Symmetric and Skew Matrices 96
1. Involutions of 2tt» 96
2. Involutions determined by ft 97
3. The 17 equivalence of matrices 99
4. Elementary T transfonnations 100
5. The first reduction of T symmetric matrices 103
6. The Kronecker reduction 104
7. Skew matrices 108
8. Matrices over an algebraically closed field 109
9. Ordered fields 110
10. Formulation of the classical theory 112
11. Reduction of real symmetric and complex Hennitian matrices . 114
12. Positive definite matrices 116
13. Orthogonal ^ equivalence 117
14. Orthogonal equivalence of real skew matrices 120
15. Forms 124
VI. Finite Groups 126
1. Subsets 126
2. Subgroups 127
3. Cyclic groups 128
4. Conjugate groups 131
5. Quotient groups 132
6. Composition series 134
7. The Sylow theorems 138
8. Permutation groups 138
9. The order of a permutation 140
10. A composition series of ©» 142
11. Finite permutation groups 144
VII. Fields over 8 146
1. Symmetric functions 146
2. Fields over 5 151
3. Simple extensions of g 152
4. Algebraic fields over g 153
TABLE OF CONTENTS xi
CHAPTXR PiOI
5. The root field of an equation 156
6. Conjugate fields 160
7. Composites 161
8. Fields of characteristic p 162
9. Separable equations 164
10. Finite fields 166
11. Separable fields 168
12. A lemma on infinite fields 170
VIII. The Galois Theort 172
1. Normal fields 172
2. The automorphisms of 91 172
3. The fundamental theorems of the Galois theory 173
4. Composites of 91 and 3 180
5. Relative traces and norms 181
6. The Galois group of an equation 183
7. Some properties of ®0 184
8. Equations with a prescribed group 184
9. Direct products 185
10. Metacyclic fields over g 186
11. Cyclotomic fields 187
12. Solution by radicals 189
IX. Cyclic Fields 192
1. The structure problem 192
2. Generating automorphisms and elements of 3 over g . . . . 193
3. Necessary conditions for fields of characteristic p 194
4. The norm and trace theorems 199
5. Sufficiency theorems for fields of characteristic p 203
6. Cyclic fields of degree p4 over a field containing a primitive pth
root of unity 206
7. Adjunction of f 209
X. Algebras of Matrices 217
1. Algebras of order n over g 217
2. The regular representation of an algebra 217
3. The characteristic and minimum functions, scalar extension . . 222
4. Direct products and sums 226
5. Direct products of total matric algebras 227
6. The degree of a total matric algebra 232
7. Quadrate algebras 233
8. Matrices with separable characteristic equations 236
9. The cyclic representation of a total matric algebra 239
10. Cyclic algebras 242
11. The matrices commutative with a subfield of 2K« 244
12. The polynomial algebra SU] 245
xii TABLE OF CONTENTS
CHAPTER FAQB
XI. Introduction to the Transcendental Theory of Fields . . . 251
1. Archimedean ordered fields 251
2. Elements of the theory of ideals 252
3. Ideals in a commutative ring 254
4. Fields with a valuation 255
5. Regular sequences of g 257
6. The derived field of g 260
7. Certain conventions about derived fields 261
8. Limits in a complete field 263
9. The case 8 = X 263
10. The real number system 266
11. A valuation of g« 267
12. Algebraic extensions 269
13. Quadratic extensions of a complete field 272
XII. Valuation Functions 277
1. Introduction 277
2. Lemmas from analysis 277
3. The types of valuations 281
4. Archimedean valuations of SR 282
5. Archimedean valued fields 284
6. Non archimedean valuations of SR 289
7. The p adic number fields SR, 290
8. The series representation of p adic numbers 292
9. Algebraic extensions of an integral domain 294
10. Algebraic extensions of p adic number fields 296
11. The valuation theory of St over SRP 298
12. Ideals in algebraic number fields 301
13. Fields of ? adic numbers 303
14. The literature 305
Glossary 307
Index 313
|
| adam_txt |
TABLE OF CONTENTS
CHAPTER PAGE
I. Groups and Rings 1
1. Introduction 1
2. Sets 1
3. Correspondences 2
4. Integers 5
5. Groups 7
6. Equivalence, subgroups 10
7. Transformation groups 12
8. Rings 13
9. Some properties of rings 15
10. Linear sets over a ring 16
11. Sequences 16
12. Polynomials over S 17
II. Rings with a Unity Element 21
1. The ring g 21
2. The characteristic of a ring 22
3. The ring ?l[x] 24
4. Integral domains and fields 26
5. Equivalence, subsystems 29
6. Divisibility 30
7. The integral domain %[x] 32
8. Unique factorization domains 34
9. The domain 3[x] 36
10. The results of the text 39
11. Linear sets over g 40
12. Forms 43
13. Algebras over g 43
III. Matrices 46
1. Rectangular matrices 46
2. Multiplication of matrices 48
3. Determinants 52
4. The rank of matrices 53
5. The algebra 5KB . .* 55
6. Elementary transformations over 3 59
7. Equivalence of matrices in a field 8 62
8. Linear equations 67
9. Bilinear forms 70
10. Equivalence in g[X] 70
ix
x TABLE OF CONTENTS
OOAPTSB Ti.O*
IV. SIMILARITY OF SQUARE MATRICES 75
1. linear transformations and similar matrices 75
2. The meaning of canonical forms 76
3. Scalar polynomials 78
4. Canonical forms 80
5. Elementary divisors 82
6. Nilpotent matrices 87
7. Idempotent matrices 88
8. The automorphisms of 2K. 89
9. Another basis of 2R» 91
V. Symmetric and Skew Matrices 96
1. Involutions of 2tt» 96
2. Involutions determined by ft 97
3. The 17 equivalence of matrices 99
4. Elementary T transfonnations 100
5. The first reduction of T symmetric matrices 103
6. The Kronecker reduction 104
7. Skew matrices 108
8. Matrices over an algebraically closed field 109
9. Ordered fields 110
10. Formulation of the classical theory 112
11. Reduction of real symmetric and complex Hennitian matrices . 114
12. Positive definite matrices 116
13. Orthogonal ^ equivalence 117
14. Orthogonal equivalence of real skew matrices 120
15. Forms 124
VI. Finite Groups 126
1. Subsets 126
2. Subgroups 127
3. Cyclic groups 128
4. Conjugate groups 131
5. Quotient groups 132
6. Composition series 134
7. The Sylow theorems 138
8. Permutation groups 138
9. The order of a permutation 140
10. A composition series of ©» 142
11. Finite permutation groups 144
VII. Fields over 8 146
1. Symmetric functions 146
2. Fields over 5 151
3. Simple extensions of g 152
4. Algebraic fields over g 153
TABLE OF CONTENTS xi
CHAPTXR PiOI
5. The root field of an equation 156
6. Conjugate fields 160
7. Composites 161
8. Fields of characteristic p 162
9. Separable equations 164
10. Finite fields 166
11. Separable fields 168
12. A lemma on infinite fields 170
VIII. The Galois Theort 172
1. Normal fields 172
2. The automorphisms of 91 172
3. The fundamental theorems of the Galois theory 173
4. Composites of 91 and 3 180
5. Relative traces and norms 181
6. The Galois group of an equation 183
7. Some properties of ®0 184
8. Equations with a prescribed group 184
9. Direct products 185
10. Metacyclic fields over g 186
11. Cyclotomic fields 187
12. Solution by radicals 189
IX. Cyclic Fields 192
1. The structure problem 192
2. Generating automorphisms and elements of 3 over g . . . . 193
3. Necessary conditions for fields of characteristic p 194
4. The norm and trace theorems 199
5. Sufficiency theorems for fields of characteristic p 203
6. Cyclic fields of degree p4 over a field containing a primitive pth
root of unity 206
7. Adjunction of f 209
X. Algebras of Matrices 217
1. Algebras of order n over g 217
2. The regular representation of an algebra 217
3. The characteristic and minimum functions, scalar extension . . 222
4. Direct products and sums 226
5. Direct products of total matric algebras 227
6. The degree of a total matric algebra 232
7. Quadrate algebras 233
8. Matrices with separable characteristic equations 236
9. The cyclic representation of a total matric algebra 239
10. Cyclic algebras 242
11. The matrices commutative with a subfield of 2K« 244
12. The polynomial algebra SU] 245
xii TABLE OF CONTENTS
CHAPTER FAQB
XI. Introduction to the Transcendental Theory of Fields . . . 251
1. Archimedean ordered fields 251
2. Elements of the theory of ideals 252
3. Ideals in a commutative ring 254
4. Fields with a valuation 255
5. Regular sequences of g 257
6. The derived field of g 260
7. Certain conventions about derived fields 261
8. Limits in a complete field 263
9. The case 8 = X 263
10. The real number system 266
11. A valuation of g« 267
12. Algebraic extensions 269
13. Quadratic extensions of a complete field 272
XII. Valuation Functions 277
1. Introduction 277
2. Lemmas from analysis 277
3. The types of valuations 281
4. Archimedean valuations of SR 282
5. Archimedean valued fields 284
6. Non archimedean valuations of SR 289
7. The p adic number fields SR, 290
8. The series representation of p adic numbers 292
9. Algebraic extensions of an integral domain 294
10. Algebraic extensions of p adic number fields 296
11. The valuation theory of St over SRP 298
12. Ideals in algebraic number fields 301
13. Fields of ? adic numbers 303
14. The literature 305
Glossary 307
Index 313 |
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| author | Albert, A. Adrian 1905-1972 |
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| author_facet | Albert, A. Adrian 1905-1972 |
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| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 10. impr. |
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| illustrated | Not Illustrated |
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| indexdate | 2024-07-09T20:47:37Z |
| institution | BVB |
| language | English |
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| physical | XII, 319 S. |
| publishDate | 1965 |
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| spelling | Albert, A. Adrian 1905-1972 Verfasser (DE-588)122621336 aut Modern higher algebra by A. Adrian Albert 10. impr. Chicago [u.a.] Univ. of Chicago Press 1965 XII, 319 S. txt rdacontent n rdamedia nc rdacarrier Algebra (DE-588)4001156-2 gnd rswk-swf Algebra (DE-588)4001156-2 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015144366&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Albert, A. Adrian 1905-1972 Modern higher algebra Algebra (DE-588)4001156-2 gnd |
| subject_GND | (DE-588)4001156-2 |
| title | Modern higher algebra |
| title_auth | Modern higher algebra |
| title_exact_search | Modern higher algebra |
| title_exact_search_txtP | Modern higher algebra |
| title_full | Modern higher algebra by A. Adrian Albert |
| title_fullStr | Modern higher algebra by A. Adrian Albert |
| title_full_unstemmed | Modern higher algebra by A. Adrian Albert |
| title_short | Modern higher algebra |
| title_sort | modern higher algebra |
| topic | Algebra (DE-588)4001156-2 gnd |
| topic_facet | Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015144366&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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