Galois' theory of algebraic equations:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Singapore [u.a.]
World Scientific
2004
|
| Ausgabe: | Reprinted |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIII, 333 S. |
| ISBN: | 9810245416 |
Internformat
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Datensatz im Suchindex
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| adam_text | GALOIS THEORY OF ALGEBRAIC EQUATIONS JEAN-PIERRE TIGNOL UNIVERSITE
CATHOLIQUE DE LOUVAIN, BELGIUM 1I§5 WORLD SCIENTIFIC WL SINQAPORE»
NEWJERSEY L SINGAPORE * NEW JERSEY LONDON * HONG KONG CONTENTS PREFACE
VII CHAPTER 1 QUADRATIC EQUATIONS 1 1.1 INTRODUCTION 1 1.2 BABYLONIAN
ALGEBRA 2 1.3 GREEK ALGEBRA 5 1.4 ARABIC ALGEBRA 9 CHAPTER 2 CUBIC
EQUATIONS 13 2.1 PRIORITY DISPUTES ON THE SOLUTION OF CUBIC EQUATIONS 13
2.2 CARDANO S FORMULA 15 2.3 DEVELOPMENTS ARISING FROM CARDANO S FORMULA
16 CHAPTER 3 QUARTIC EQUATIONS 21 3.1 THE UNNATURALNESS OF QUARTIC
EQUATIONS 21 3.2 FERRARI S METHOD 22 CHAPTER 4 THE CREATION OF
POLYNOMIALS 25 4.1 THE RISE OF SYMBOLIC ALGEBRA 25 4.1.1 L ARITHMETIQUE
26 4.1.2 IN ARTEM ANALYTICEM ISAGOGE 29 4.2 RELATIONS BETWEEN ROOTS AND
COEFFICIENTS 30 CHAPTER 5 A MODERN APPROACH TO POLYNOMIALS 41 5.1
DEFMITIONS 41 5.2 EUCLIDEAN DIVISION 43 XI XII CONTENTS 5.3 IRREDUCIBLE
POLYNOMIALS 48 5.4 ROOTS 50 5.5 MULTIPLE ROOTS AND DERIVATIVES 53 5.6
COMMON ROOTS OF TWO POLYNOMIALS 56 APPENDIX: DECOMPOSITION OF RATIONAL
FRACTIONS IN SUMS OF PARTIAL FRACTIONS . 58 CHAPTER 6 ALTERNATIVE
METHODS FOR CUBIC AND QUARTIC EQUATIONS 61 6.1 VIETE ON CUBIC EQUATIONS
61 6.1.1 TRIGONOMETRIE SOLUTION FOR THE IRREDUCIBLE CASE 61 6.1.2
ALGEBRAIC SOLUTION FOR THE GENERAL CASE 62 6.2 DESCARTES ON QUARTIC
EQUATIONS 64 6.3 RATIONAL SOLUTIONS FOR EQUATIONS WITH RATIONAL
COEFFICIENTS 65 6.4 TSCHIRNHAUS METHOD 67 CHAPTER 7 ROOTS OFUNITY 73
7.1 INTRODUCTION 73 7.2 THE ORIGINOF DEMOIVRE S FORMULA 74 7.3 THE ROOTS
OF UNITY 81 7.4 PRIMITIVE ROOTS AND CYCLOTOMIC POLYNOMIALS 86 APPENDIX:
LEIBNIZ AND NEWTON ON THE SUMMATION OF SERIES 92 EXERCISES 94 CHAPTER 8
SYMMETRIE FUNCTIONS 97 8.1 INTRODUCTION 97 8.2 WARING S METHOD 100 8.3
THE DISCRIMINANT 106 APPENDIX: EULER S SUMMATION OF THE SERIES OF
RECIPROCALS OF PERFECT SQUARES 110 EXERCISES 112 CHAPTER 9 THE
FUNDAMENTAL THEOREM OF ALGEBRA 115 9.1 INTRODUCTION 115 9.2 GIRARD S
THEOREM 116 9.3 PROOF OF THE FUNDAMENTAL THEOREM 119 CHAPTER 10 LAGRANGE
123 10.1 THE THEORY OF EQUATIONS COMES OF AGE 123 10.2 LAGRANGE S
OBSERVATIONS ON PREVIOUSLY KNOWN METHODS 127 10.3 FIRST RESULTS OF GROUP
THEORY AND GALOIS THEORY 138 EXERCISES 150 CONTENTS XIII CHAPTER 11
VANDERMONDE 153 11.1 INTRODUCTION 153 11.2 THE SOLUTION OF GENERAL
EQUATIONS 154 11.3 CYCLOTOMIC EQUATIONS 158 EXERCISES 164 CHAPTER 12
GAUSS ON CYCLOTOMIC EQUATIONS 167 12.1 INTRODUCTION 167 12.2
NUMBER-THEORETIC PRELIMINARIES 168 12.3 IRREDUCIBILITY OF THE CYCLOTOMIC
POLYNOMIALS OF PRIME INDEX 175 12.4 THE PERIODS OF CYCLOTOMIC EQUATIONS
182 12.5 SOLVABILITY BY RADICALS 192 12.6 IRREDUCIBILITY OF THE
CYCLOTOMIC POLYNOMIALS 196 APPENDIX: RULER AND COMPASS CONSTRUCTION OF
REGULAER POLYGONS 200 EXERCISES 206 CHAPTER 13 RUFFINI AND ABEL ON
GENERAL EQUATIONS 209 13.1 INTRODUCTION 209 13.2 RADICAL EXTENSIONS 212
13.3 ABEL S THEOREM ON NATURAL IRRATIONALITIES 218 13.4 PROOF OF THE
UNSOLVABILITY OF GENERAL EQUATIONS OF DEGREE HIGHER THAN 4 225 EXERCISES
227 CHAPTER 14 GALOIS 231 14.1 INTRODUCTION 231 14.2 THE GALOIS GROUP OF
AN EQUATION 235 14.3 THE GALOIS GROUP UNDER FIELD EXTENSION 254 14.4
SOLVABILITY BY RADICALS 264 14.5 APPLICATIONS 281 APPENDIX: GALOIS
DESCRIPTION OF GROUPS OF PERMUTATIONS 295 EXERCISES 301 CHAPTER 15
EPILOGUE 303 APPENDIX: THE FUNDAMENTAL THEOREM OF GALOIS THEORY 307
EXERCISES 315 SELECTED SOLUTIONS 317 BIBLIOGRAPHY 325 INDEX 331
|
| adam_txt |
GALOIS'THEORY OF ALGEBRAIC EQUATIONS JEAN-PIERRE TIGNOL UNIVERSITE
CATHOLIQUE DE LOUVAIN, BELGIUM 1I§5 WORLD SCIENTIFIC WL SINQAPORE»
NEWJERSEY L SINGAPORE * NEW JERSEY LONDON * HONG KONG CONTENTS PREFACE
VII CHAPTER 1 QUADRATIC EQUATIONS 1 1.1 INTRODUCTION 1 1.2 BABYLONIAN
ALGEBRA 2 1.3 GREEK ALGEBRA 5 1.4 ARABIC ALGEBRA 9 CHAPTER 2 CUBIC
EQUATIONS 13 2.1 PRIORITY DISPUTES ON THE SOLUTION OF CUBIC EQUATIONS 13
2.2 CARDANO'S FORMULA 15 2.3 DEVELOPMENTS ARISING FROM CARDANO'S FORMULA
16 CHAPTER 3 QUARTIC EQUATIONS 21 3.1 THE UNNATURALNESS OF QUARTIC
EQUATIONS 21 3.2 FERRARI'S METHOD 22 CHAPTER 4 THE CREATION OF
POLYNOMIALS 25 4.1 THE RISE OF SYMBOLIC ALGEBRA 25 4.1.1 L'ARITHMETIQUE
26 4.1.2 IN ARTEM ANALYTICEM ISAGOGE 29 4.2 RELATIONS BETWEEN ROOTS AND
COEFFICIENTS 30 CHAPTER 5 A MODERN APPROACH TO POLYNOMIALS 41 5.1
DEFMITIONS 41 5.2 EUCLIDEAN DIVISION 43 XI XII CONTENTS 5.3 IRREDUCIBLE
POLYNOMIALS 48 5.4 ROOTS 50 5.5 MULTIPLE ROOTS AND DERIVATIVES 53 5.6
COMMON ROOTS OF TWO POLYNOMIALS 56 APPENDIX: DECOMPOSITION OF RATIONAL
FRACTIONS IN SUMS OF PARTIAL FRACTIONS . 58 CHAPTER 6 ALTERNATIVE
METHODS FOR CUBIC AND QUARTIC EQUATIONS 61 6.1 VIETE ON CUBIC EQUATIONS
61 6.1.1 TRIGONOMETRIE SOLUTION FOR THE IRREDUCIBLE CASE 61 6.1.2
ALGEBRAIC SOLUTION FOR THE GENERAL CASE 62 6.2 DESCARTES ON QUARTIC
EQUATIONS 64 6.3 RATIONAL SOLUTIONS FOR EQUATIONS WITH RATIONAL
COEFFICIENTS 65 6.4 TSCHIRNHAUS' METHOD 67 CHAPTER 7 ROOTS OFUNITY 73
7.1 INTRODUCTION 73 7.2 THE ORIGINOF DEMOIVRE'S FORMULA 74 7.3 THE ROOTS
OF UNITY 81 7.4 PRIMITIVE ROOTS AND CYCLOTOMIC POLYNOMIALS 86 APPENDIX:
LEIBNIZ AND NEWTON ON THE SUMMATION OF SERIES 92 EXERCISES 94 CHAPTER 8
SYMMETRIE FUNCTIONS 97 8.1 INTRODUCTION 97 8.2 WARING'S METHOD 100 8.3
THE DISCRIMINANT 106 APPENDIX: EULER'S SUMMATION OF THE SERIES OF
RECIPROCALS OF PERFECT SQUARES 110 EXERCISES 112 CHAPTER 9 THE
FUNDAMENTAL THEOREM OF ALGEBRA 115 9.1 INTRODUCTION 115 9.2 GIRARD'S
THEOREM 116 9.3 PROOF OF THE FUNDAMENTAL THEOREM 119 CHAPTER 10 LAGRANGE
123 10.1 THE THEORY OF EQUATIONS COMES OF AGE 123 10.2 LAGRANGE'S
OBSERVATIONS ON PREVIOUSLY KNOWN METHODS 127 10.3 FIRST RESULTS OF GROUP
THEORY AND GALOIS THEORY 138 EXERCISES 150 CONTENTS XIII CHAPTER 11
VANDERMONDE 153 11.1 INTRODUCTION 153 11.2 THE SOLUTION OF GENERAL
EQUATIONS 154 11.3 CYCLOTOMIC EQUATIONS 158 EXERCISES 164 CHAPTER 12
GAUSS ON CYCLOTOMIC EQUATIONS 167 12.1 INTRODUCTION 167 12.2
NUMBER-THEORETIC PRELIMINARIES 168 12.3 IRREDUCIBILITY OF THE CYCLOTOMIC
POLYNOMIALS OF PRIME INDEX 175 12.4 THE PERIODS OF CYCLOTOMIC EQUATIONS
182 12.5 SOLVABILITY BY RADICALS 192 12.6 IRREDUCIBILITY OF THE
CYCLOTOMIC POLYNOMIALS 196 APPENDIX: RULER AND COMPASS CONSTRUCTION OF
REGULAER POLYGONS 200 EXERCISES 206 CHAPTER 13 RUFFINI AND ABEL ON
GENERAL EQUATIONS 209 13.1 INTRODUCTION 209 13.2 RADICAL EXTENSIONS 212
13.3 ABEL'S THEOREM ON NATURAL IRRATIONALITIES 218 13.4 PROOF OF THE
UNSOLVABILITY OF GENERAL EQUATIONS OF DEGREE HIGHER THAN 4 225 EXERCISES
227 CHAPTER 14 GALOIS 231 14.1 INTRODUCTION 231 14.2 THE GALOIS GROUP OF
AN EQUATION 235 14.3 THE GALOIS GROUP UNDER FIELD EXTENSION 254 14.4
SOLVABILITY BY RADICALS 264 14.5 APPLICATIONS 281 APPENDIX: GALOIS'
DESCRIPTION OF GROUPS OF PERMUTATIONS 295 EXERCISES 301 CHAPTER 15
EPILOGUE 303 APPENDIX: THE FUNDAMENTAL THEOREM OF GALOIS THEORY 307
EXERCISES 315 SELECTED SOLUTIONS 317 BIBLIOGRAPHY 325 INDEX 331 |
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| author | Tignol, Jean-Pierre 1954- |
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| dewey-ones | 512 - Algebra |
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| dewey-sort | 3512 13 |
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| discipline | Mathematik |
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T15:24:54Z |
| indexdate | 2024-07-09T20:42:37Z |
| institution | BVB |
| isbn | 9810245416 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014939866 |
| oclc_num | 255298650 |
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| owner | DE-384 DE-824 |
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| physical | XIII, 333 S. |
| publishDate | 2004 |
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| publisher | World Scientific |
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| spelling | Tignol, Jean-Pierre 1954- Verfasser (DE-588)1018527419 aut Leçons sur la théorie des équations Galois' theory of algebraic equations Jean-Pierre Tignol Reprinted Singapore [u.a.] World Scientific 2004 XIII, 333 S. txt rdacontent n rdamedia nc rdacarrier Algebraische Gleichung - Galois-Theorie Equations, Theory of Galois theory Algebraische Gleichung (DE-588)4001162-8 gnd rswk-swf Galois-Theorie (DE-588)4155901-0 gnd rswk-swf Algebraische Gleichung (DE-588)4001162-8 s Galois-Theorie (DE-588)4155901-0 s DE-604 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014939866&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Tignol, Jean-Pierre 1954- Galois' theory of algebraic equations Algebraische Gleichung - Galois-Theorie Equations, Theory of Galois theory Algebraische Gleichung (DE-588)4001162-8 gnd Galois-Theorie (DE-588)4155901-0 gnd |
| subject_GND | (DE-588)4001162-8 (DE-588)4155901-0 |
| title | Galois' theory of algebraic equations |
| title_alt | Leçons sur la théorie des équations |
| title_auth | Galois' theory of algebraic equations |
| title_exact_search | Galois' theory of algebraic equations |
| title_exact_search_txtP | Galois' theory of algebraic equations |
| title_full | Galois' theory of algebraic equations Jean-Pierre Tignol |
| title_fullStr | Galois' theory of algebraic equations Jean-Pierre Tignol |
| title_full_unstemmed | Galois' theory of algebraic equations Jean-Pierre Tignol |
| title_short | Galois' theory of algebraic equations |
| title_sort | galois theory of algebraic equations |
| topic | Algebraische Gleichung - Galois-Theorie Equations, Theory of Galois theory Algebraische Gleichung (DE-588)4001162-8 gnd Galois-Theorie (DE-588)4155901-0 gnd |
| topic_facet | Algebraische Gleichung - Galois-Theorie Equations, Theory of Galois theory Algebraische Gleichung Galois-Theorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014939866&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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