Galois' theory of algebraic equations:
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo
World Scientific
[2016]
|
| Ausgabe: | Second edition |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | xvi, 308 Seiten Diagramme |
| ISBN: | 9789814704694 |
Internformat
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| 240 | 1 | 0 | |a Leçons sur la théorie des équations |
| 245 | 1 | 0 | |a Galois' theory of algebraic equations |c Jean-Pierre Tignol, Université Catholique de Louvain, Belgium |
| 250 | |a Second edition | ||
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| 264 | 4 | |c © 2016 | |
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Datensatz im Suchindex
| _version_ | 1804176171778179072 |
|---|---|
| adam_text | Titel: Galois theory of algebraic equations
Autor: Tignol, Jean-Pierre
Jahr: 2016
Contents Preface, to the Second Edition Preface to the First Edition (2001) 1. Quadratic Equations 1.1 Babylonian algebra...................... 1.2 Greek algebra......................... 1.3 Arabic algebra ........................ 2. Cubic Equations 2.1 Priority disputes on the solution of cubic equations . . . . 2.2 Cardano’s formula...................... 2.3 Developments arising from Cardano’s formula........ 3. Quartic Equations 3.1 The unnaturalness of quartic equations........... 3.2 Ferrari s method ....................... 4. The Creation of Polynomials 4.1 The rise of symbolic algebra.................. 4.1.1 L’Aritlnnetique.................... 4.1.2 In Artem Analyticem Isagoge............ 4.2 Relations between roots and coefficients.......... 5. vii ix 1 2 5 9 13 13 15 16 21 21 22 25 25 26 29 30 A Modern Approach to Polynomials 5.1 Definitions........... 5.2 Euclidean division ...... 41 41 43
XIV Contents 5.3 Irreducible polynomials.................... 48 5.4 Roots ............................. 51 5.5 Multiple roots and derivatives................ 53 5.6 Common roots of two polynomials ............. 56 Appendix: Decomposition of rational functions into sums of partial fractions........................ 60 6. Alternative Methods for Cubic and Quartic Equations 63 6.1 Viète on cubic equations................... 63 6.1.1 Trigonometric solution for the irreducible case . . 63 6.1.2 Algebraic solution for the general case....... 64 6.2 Descartes on quartic equations ............... 66 6.3 Rational solutions for equations with rational coefficients . 67 6.4 Tschirnhaus’ method..................... 68 7. Roots of Unity 73 7.1 The origins of de Moivre’s formula............. 73 7.2 The roots of unity ...................... 80 7.3 Primitive roots and cyclotomie polynomials........ 85 Appendix: Leibniz and Newton on the summation of series ... 89 Exercises............................... 90 8. Symmetric Functions 93 8.1 YVaring s method....................... 96 8.2 The discriminant....................... 101 Appendix: Euler’s summation of the series of reciprocals of perfect squares........................ 105 Exercise s............................... 107 9. The Fundamental Theorem of Algebra 109 9.1 Girard s theorem....................... 110 9.2 Proof of the fundamental theorem.............. 113 10. Lagrange 117 10.1 The theory of equations comes of age............ 117 10.2 Lagrange s observations on previously known methods . . 121 10.3 First, results of group theory and Galois theory...... 131 Exercise s............................... 142
Contents xv 11. Vandermonde 143 11.1 The solution of general equations .............. 144 11.2 Cyclotomie equations ..................... 148 Exercises............................... 154 12. Gauss on Cyclotomie Equations 155 12.1 Number-theoretic preliminaries ............... 150 12.2 Irreducibility of the cyclotomie polynomials of prime index 162 12.3 The periods of cyclotomie equations ............ 169 12.4 Solvability by radicals.................... 178 12.5 Irreducibility of the cyclotomie polynomials........ 182 Appendix: Ruler and compass construction of regular polygons 185 Exercises............................... 192 13. Ruffini and Abel on General Equations 193 13.1 Radical extensions...................... 195 13.2 Abel s theorem on natural irrationalities.......... 203 13.3 Proof of the unsolvability of general equations of degree higher than 4......................... 209 Exercises............................... 211 14. Galois 215 14.1 Arrangements and permutations .............. 220 14.2 The Galois group of an equation .............. 225 14.3 The Galois group under base field extension........ 236 14.4 Solvability by radicals .................... 246 14.5 Applications.......................... 256 14.5.1 Irreducible equations of prime degree....... 256 14.5.2 Abelian equations.................. 265 Exercises............................... 268 15. Epilogue 271 Appendix 1: The fundamental theorem of Galois theory .... 274 Appendix 2: Galois theory à la Grothendieck .......... 283 Étale algebras......................... 283 Galois algebras........................ 285 Galois groups......................... 287 Exercises............................... 290
XVI Contents Selected Solut,ions Bibliography Index 291 299 305
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| discipline | Mathematik |
| edition | Second edition |
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| id | DE-604.BV043523832 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:27:57Z |
| institution | BVB |
| isbn | 9789814704694 |
| language | English |
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| physical | xvi, 308 Seiten Diagramme |
| publishDate | 2016 |
| publishDateSearch | 2016 |
| publishDateSort | 2016 |
| publisher | World Scientific |
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| spelling | Tignol, Jean-Pierre 1954- (DE-588)1018527419 aut Leçons sur la théorie des équations Galois' theory of algebraic equations Jean-Pierre Tignol, Université Catholique de Louvain, Belgium Second edition New Jersey ; London ; Singapore ; Beijing ; Shanghai ; Hong Kong ; Taipei ; Chennai ; Tokyo World Scientific [2016] © 2016 xvi, 308 Seiten Diagramme 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 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028939692&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_full | Galois' theory of algebraic equations Jean-Pierre Tignol, Université Catholique de Louvain, Belgium |
| title_fullStr | Galois' theory of algebraic equations Jean-Pierre Tignol, Université Catholique de Louvain, Belgium |
| title_full_unstemmed | Galois' theory of algebraic equations Jean-Pierre Tignol, Université Catholique de Louvain, Belgium |
| 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=028939692&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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