A history of abstract algebra:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boston [u.a.]
Birkhäuser
2007
|
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | XIII, 168 S. Ill. 23 cm |
| ISBN: | 9780817646844 |
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Datensatz im Suchindex
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CONTENTS PREFACE
. XI PERMISSIONS
. XV
HISTORYOFCLASSICALALGEBRA . 1 1.1
EARLYROOTS. 1 1.2 THE
GREEKS . 2 1.3
AL-KHWARIZMI . 3 1.4 CUBIC
AND QUARTICEQUATIONS. 5 1.5 THE
CUBICAND COMPLEX NUMBERS . 7 1.6
ALGEBRAICNOTATION: VIETEAND DESCARTES . 8 1.7 THE
THEORYOFEQUATIONSAND THEFUNDAMENTAL THEOREM OFALGEBRA . 10 1.8
SYMBOLICALALGEBRA. 13 REFERENCES
. 14 2
HISTORYOFGROUP THEORY . 17 2.1
SOURCESOFGROUP THEORY. 17 2.1.1
CLASSICAL ALGEBRA . 18 2.1.2 NUMBER
THEORY . 19 2.1.3 GEOMETRY
. 20 2.1.4 ANALYSIS
. 21 2.2 DEVELOPMENT
OF"SPECIALIZED" THEORIESOFGROUPS . 22 2.2.1
PERMUTATIONGROUPS . 22 2.2.2
ABELIANGROUPS . 26 2.2.3
TRANSFORMATIONGROUPS . 28 2.3 EMERGENCE
OFABSTRACTION INGROUP THEORY . 30 2.4
CONSOLIDATIONOFTHEABSTRACT GROUP CONCEPT;DAWN OF ABSTRACT GROUP
THEORY. 33 2.5
DIVERGENCEOFDEVELOPMENTSINGROUP THEORY . 35
REFERENCES . 38 VIII
CONTENTS 3 HISTORYOFRING THEORY .
41 3.1 NONCOMMUTATIVE RINGTHEORY . 42
3.1.1 EXAMPLES OFHYPERCOMPLEX NUMBER SYSTEMS . 42 3.1.2
CLASSIFICATION . 43 3.1.3
STRUCTURE . 45 3.2 COMMUTATIVE
RINGTHEORY . 47 3.2.1 ALGEBRAICNUMBER
THEORY . 48 3.2.2 ALGEBRAICGEOMETRY
. 54 3.2.3 INVARIANT THEORY
. 57 3.3 THE ABSTRACT DEFINITION OFA
RING. 58 3.4 EMMY NOETHERAND EMIL ARTIN
. 59 3.5 EPILOGUE
. 60 REFERENCES
. 60 4
HISTORYOFFIELDTHEORY . 63 4.1
GALOISTHEORY . 63 4.2
ALGEBRAICNUMBER THEORY . 64 4.2.1
DEDEKIND'SIDEAS . 65 4.2.2
KRONECKER'SIDEAS . 67 4.2.3 DEDEKIND
VS KRONECKER . 68 4.3 ALGEBRAICGEOMETRY
. 68 4.3.1
FIELDSOFALGEBRAICFUNCTIONS . 68 4.3.2
FIELDSOFRATIONALFUNCTIONS . 70 4.4 CONGRUENCES
. 70 4.5
SYMBOLICALALGEBRA. 71 4.6 THE
ABSTRACT DEFINITION OFA FIELD . 71 4.7
HENSEL'SP-ADICNUMBERS . 73 4.8
STEINITZ . 74 4.9 A
GLANCEAHEAD . 76 REFERENCES
. 77 5
HISTORYOFLINEAR ALGEBRA . 79 5.1
LINEAREQUATIONS. 79 5.2
DETERMINANTS . 81 5.3
MATRICESAND LINEARTRANSFORMATIONS. 82 5.4
LINEARINDEPENDENCE,BASIS,AND DIMENSION . 84 5.5
VECTORSPACES. 86 REFERENCES
. 89 6 EMMY NOETHER
AND THEADVENT OFABSTRACTALGEBRA . 91 6.1 INVARIANT THEORY
. 92 6.2 COMMUTATIVE
ALGEBRA. 94 6.3 NONCOMMUTATIVE
ALGEBRAAND REPRESENTATION THEORY. 97 6.4
APPLICATIONSOFNONCOMMUTATIVE TOCOMMUTATIVE ALGEBRA. 98 6.5
NOETHER'SLEGACY . 99
REFERENCES . 101
CONTENTS IX 7A COURSE INABSTRACTALGEBRA INSPIREDBY HISTORY
. 103 PROBLEM I:WHY IS(-1)(-1) = 1?
. 104 PROBLEM II:WHAT
ARETHEINTEGERSOLUTIONS OFX2 +2= Y3?. 105 PROBLEM III:CAN WE
TRISECT A 60 ANGLEUSINGONLYSTRAIGHTEDGE AND COMPASS?
. 106 PROBLEM IV:CAN WE
SOLVEX5 - 6X +3=0 BY RADICALS?. 107 PROBLEM V: "PAPA,CAN
YOU MULTIPLYTRIPLES? " . 108 GENERALREMARKSON
THECOURSE . 109 REFERENCES
. 110 8
BIOGRAPHIESOFSELECTEDMATHEMATICIANS . 113 8.1
ARTHURCAYLEY(1821-1895) . 113 8.1.1
INVARIANTS . 115 8.1.2 GROUPS
. 116 8.1.3 MATRICES
. 117 8.1.4 GEOMETRY
. 118 8.1.5 CONCLUSION
. 119 REFERENCES
. 120 8.2
RICHARDDEDEKIND(1831-1916) . 121 8.2.1
ALGEBRAICNUMBERS . 124 8.2.2
REALNUMBERS . 126 8.2.3
NATURALNUMBERS . 128 8.2.4 OTHERWORKS
. 129 8.2.5 CONCLUSION
. 131 REFERENCES
. 132 8.3
EVARISTEGALOIS(1811-1832) . 133 8.3.1
MATHEMATICS . 135 8.3.2 POLITICS
. 135 8.3.3 THE DUEL
. 137 8.3.4 TESTAMENT
. 137 8.3.5 CONCLUSION
. 138 REFERENCES
. 139 8.4
CARLFRIEDRICHGAUSS (1777-1855) . 139 8.4.1
NUMBER THEORY. 140 8.4.2
DIFFERENTIAL GEOMETRY,PROBABILITY, AND STATISTICS . 142 8.4.3
THE DIARY . 142 8.4.4
CONCLUSION . 143 REFERENCES
. 144 8.5
WILLIAMROWAN HAMILTON (1805-1865) . 144 8.5.1
OPTICS . 146 8.5.2 DYNAMICS
. 147 8.5.3 COMPLEX NUMBERS
. 149 8.5.4 FOUNDATIONSOFALGEBRA
. 150 8.5.5 QUATERNIONS
. 152 8.5.6 CONCLUSION
. 156 REFERENCES
. 156 X CONTENTS 8.6
EMMY NOETHER(1882-1935) . 157 8.6.1
EARLYYEARS . 157 8.6.2
UNIVERSITYSTUDIES . 158 8.6.3
GOTTINGEN . 159 8.6.4
NOETHERASATEACHER . 160 8.6.5 BRYN MAWR
. 161 8.6.6 CONCLUSION
. 162 REFERENCES
. 162 INDEX
. 165 |
| adam_txt |
CONTENTS PREFACE
. XI PERMISSIONS
. XV
HISTORYOFCLASSICALALGEBRA . 1 1.1
EARLYROOTS. 1 1.2 THE
GREEKS . 2 1.3
AL-KHWARIZMI . 3 1.4 CUBIC
AND QUARTICEQUATIONS. 5 1.5 THE
CUBICAND COMPLEX NUMBERS . 7 1.6
ALGEBRAICNOTATION: VIETEAND DESCARTES . 8 1.7 THE
THEORYOFEQUATIONSAND THEFUNDAMENTAL THEOREM OFALGEBRA . 10 1.8
SYMBOLICALALGEBRA. 13 REFERENCES
. 14 2
HISTORYOFGROUP THEORY . 17 2.1
SOURCESOFGROUP THEORY. 17 2.1.1
CLASSICAL ALGEBRA . 18 2.1.2 NUMBER
THEORY . 19 2.1.3 GEOMETRY
. 20 2.1.4 ANALYSIS
. 21 2.2 DEVELOPMENT
OF"SPECIALIZED" THEORIESOFGROUPS . 22 2.2.1
PERMUTATIONGROUPS . 22 2.2.2
ABELIANGROUPS . 26 2.2.3
TRANSFORMATIONGROUPS . 28 2.3 EMERGENCE
OFABSTRACTION INGROUP THEORY . 30 2.4
CONSOLIDATIONOFTHEABSTRACT GROUP CONCEPT;DAWN OF ABSTRACT GROUP
THEORY. 33 2.5
DIVERGENCEOFDEVELOPMENTSINGROUP THEORY . 35
REFERENCES . 38 VIII
CONTENTS 3 HISTORYOFRING THEORY .
41 3.1 NONCOMMUTATIVE RINGTHEORY . 42
3.1.1 EXAMPLES OFHYPERCOMPLEX NUMBER SYSTEMS . 42 3.1.2
CLASSIFICATION . 43 3.1.3
STRUCTURE . 45 3.2 COMMUTATIVE
RINGTHEORY . 47 3.2.1 ALGEBRAICNUMBER
THEORY . 48 3.2.2 ALGEBRAICGEOMETRY
. 54 3.2.3 INVARIANT THEORY
. 57 3.3 THE ABSTRACT DEFINITION OFA
RING. 58 3.4 EMMY NOETHERAND EMIL ARTIN
. 59 3.5 EPILOGUE
. 60 REFERENCES
. 60 4
HISTORYOFFIELDTHEORY . 63 4.1
GALOISTHEORY . 63 4.2
ALGEBRAICNUMBER THEORY . 64 4.2.1
DEDEKIND'SIDEAS . 65 4.2.2
KRONECKER'SIDEAS . 67 4.2.3 DEDEKIND
VS KRONECKER . 68 4.3 ALGEBRAICGEOMETRY
. 68 4.3.1
FIELDSOFALGEBRAICFUNCTIONS . 68 4.3.2
FIELDSOFRATIONALFUNCTIONS . 70 4.4 CONGRUENCES
. 70 4.5
SYMBOLICALALGEBRA. 71 4.6 THE
ABSTRACT DEFINITION OFA FIELD . 71 4.7
HENSEL'SP-ADICNUMBERS . 73 4.8
STEINITZ . 74 4.9 A
GLANCEAHEAD . 76 REFERENCES
. 77 5
HISTORYOFLINEAR ALGEBRA . 79 5.1
LINEAREQUATIONS. 79 5.2
DETERMINANTS . 81 5.3
MATRICESAND LINEARTRANSFORMATIONS. 82 5.4
LINEARINDEPENDENCE,BASIS,AND DIMENSION . 84 5.5
VECTORSPACES. 86 REFERENCES
. 89 6 EMMY NOETHER
AND THEADVENT OFABSTRACTALGEBRA . 91 6.1 INVARIANT THEORY
. 92 6.2 COMMUTATIVE
ALGEBRA. 94 6.3 NONCOMMUTATIVE
ALGEBRAAND REPRESENTATION THEORY. 97 6.4
APPLICATIONSOFNONCOMMUTATIVE TOCOMMUTATIVE ALGEBRA. 98 6.5
NOETHER'SLEGACY . 99
REFERENCES . 101
CONTENTS IX 7A COURSE INABSTRACTALGEBRA INSPIREDBY HISTORY
. 103 PROBLEM I:WHY IS(-1)(-1) = 1?
. 104 PROBLEM II:WHAT
ARETHEINTEGERSOLUTIONS OFX2 +2= Y3?. 105 PROBLEM III:CAN WE
TRISECT A 60 ANGLEUSINGONLYSTRAIGHTEDGE AND COMPASS?
. 106 PROBLEM IV:CAN WE
SOLVEX5 - 6X +3=0 BY RADICALS?. 107 PROBLEM V: "PAPA,CAN
YOU MULTIPLYTRIPLES? " . 108 GENERALREMARKSON
THECOURSE . 109 REFERENCES
. 110 8
BIOGRAPHIESOFSELECTEDMATHEMATICIANS . 113 8.1
ARTHURCAYLEY(1821-1895) . 113 8.1.1
INVARIANTS . 115 8.1.2 GROUPS
. 116 8.1.3 MATRICES
. 117 8.1.4 GEOMETRY
. 118 8.1.5 CONCLUSION
. 119 REFERENCES
. 120 8.2
RICHARDDEDEKIND(1831-1916) . 121 8.2.1
ALGEBRAICNUMBERS . 124 8.2.2
REALNUMBERS . 126 8.2.3
NATURALNUMBERS . 128 8.2.4 OTHERWORKS
. 129 8.2.5 CONCLUSION
. 131 REFERENCES
. 132 8.3
EVARISTEGALOIS(1811-1832) . 133 8.3.1
MATHEMATICS . 135 8.3.2 POLITICS
. 135 8.3.3 THE DUEL
. 137 8.3.4 TESTAMENT
. 137 8.3.5 CONCLUSION
. 138 REFERENCES
. 139 8.4
CARLFRIEDRICHGAUSS (1777-1855) . 139 8.4.1
NUMBER THEORY. 140 8.4.2
DIFFERENTIAL GEOMETRY,PROBABILITY, AND STATISTICS . 142 8.4.3
THE DIARY . 142 8.4.4
CONCLUSION . 143 REFERENCES
. 144 8.5
WILLIAMROWAN HAMILTON (1805-1865) . 144 8.5.1
OPTICS . 146 8.5.2 DYNAMICS
. 147 8.5.3 COMPLEX NUMBERS
. 149 8.5.4 FOUNDATIONSOFALGEBRA
. 150 8.5.5 QUATERNIONS
. 152 8.5.6 CONCLUSION
. 156 REFERENCES
. 156 X CONTENTS 8.6
EMMY NOETHER(1882-1935) . 157 8.6.1
EARLYYEARS . 157 8.6.2
UNIVERSITYSTUDIES . 158 8.6.3
GOTTINGEN . 159 8.6.4
NOETHERASATEACHER . 160 8.6.5 BRYN MAWR
. 161 8.6.6 CONCLUSION
. 162 REFERENCES
. 162 INDEX
. 165 |
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| spelling | Kleiner, Israel Verfasser (DE-588)133618234 aut A history of abstract algebra Israel Kleiner Boston [u.a.] Birkhäuser 2007 XIII, 168 S. Ill. 23 cm txt rdacontent n rdamedia nc rdacarrier Geschichte 1700-1950 gnd rswk-swf Algèbre abstraite - Histoire Geschichte Algebra, Abstract History Universelle Algebra (DE-588)4061777-4 gnd rswk-swf Universelle Algebra (DE-588)4061777-4 s Geschichte 1700-1950 z b DE-604 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2930377&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung Deutsches Museum application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015731910&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kleiner, Israel A history of abstract algebra Algèbre abstraite - Histoire Geschichte Algebra, Abstract History Universelle Algebra (DE-588)4061777-4 gnd |
| subject_GND | (DE-588)4061777-4 |
| title | A history of abstract algebra |
| title_auth | A history of abstract algebra |
| title_exact_search | A history of abstract algebra |
| title_exact_search_txtP | A history of abstract algebra |
| title_full | A history of abstract algebra Israel Kleiner |
| title_fullStr | A history of abstract algebra Israel Kleiner |
| title_full_unstemmed | A history of abstract algebra Israel Kleiner |
| title_short | A history of abstract algebra |
| title_sort | a history of abstract algebra |
| topic | Algèbre abstraite - Histoire Geschichte Algebra, Abstract History Universelle Algebra (DE-588)4061777-4 gnd |
| topic_facet | Algèbre abstraite - Histoire Geschichte Algebra, Abstract History Universelle Algebra |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=2930377&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015731910&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT kleinerisrael ahistoryofabstractalgebra |