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E-grāmata: Quadratic Reciprocity Law: A Collection of Classical Proofs

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  • Formāts: PDF+DRM
  • Izdošanas datums: 27-May-2015
  • Izdevniecība: Birkhauser Verlag AG
  • Valoda: eng
  • ISBN-13: 9783319162836
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Quadratic Reciprocity Law: A Collection of Classical ProofsPalielināt
  • Formāts: PDF+DRM
  • Izdošanas datums: 27-May-2015
  • Izdevniecība: Birkhauser Verlag AG
  • Valoda: eng
  • ISBN-13: 9783319162836
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This book is the English translation of Baumgart"s thesis on the early proofs of the quadratic reciprocity law ("Über das quadratische Reciprocitätsgesetz. Eine vergleichende Darstellung der Beweise"), first published in 1885. It is divided into two parts. The first part presents a very brief history of the development of number theory up to Legendre, as well as detailed descriptions of several early proofs of the quadratic reciprocity law. The second part highlights Baumgart"s comparisons of the principles behind these proofs. A current list of all known proofs of the quadratic reciprocity law, with complete references, is provided in the appendix.This book will appeal to all readers interested in elementary number theory and the history of number theory.

Translator"s Preface.- Baumgart"s Thesis.- Introduction.- First Part: 1. From Fermat to Legendre.- 2. Gauss"s Proof by Mathematical Induction.- 3. Proof by Reduction.- 4. Eisenstein"s Proof using Complex Analysis.- 5. Proofs using Results from Cyclotomy.- 6. Proofs based on the Theory of Quadratic Forms.- 7. The Supplementary Laws.- 8. Algorithms for Determining the Quadratic Character.- Second Part: 9. Gauss"s Proof by Induction.- 10. Proofs by Reduction.- 11. Eisenstein"s Proofs using Complex Analysis.- 12. Proofs using Results from Cyclotomy.- 13. Proofs based on the Theory of Quadratic Forms.- Final Comments.- Proofs of the Quadratic Reciprocity Law.- Author Index.- Subject Index.

"Baumgart collected and analyzed existing proofs of QRL in his 1885 thesis, translated here into English for the first time. ... Summing Up: Recommended." (D. V. Feldman, Choice, Vol. 53 (5), January, 2016)"The book has an excellent comparative discussion of many proofs along with historic notes and comments by translator. It contains a vast list of references that are updated. ... This excellent book is a necessary one for any number theorist. Every student in the field can find a lot of virgin ideas for further research as well. This book should be a good resource for mathematics historian as well." (Manouchehr Misaghian, zbMATH 1338.11003, 2016)"The book under review provides an English translation by Franz Lemmermeyer, who is an expert in both the history of mathematics and also in algebraic number theory, of this highly remarkable thesis. In particular, the many valuable comments of the translator make the reading a pleasure and accessible to mathematicians not trained in

studying the older literature." (Jörn Steuding, London Mathematical Society Newsletter, newsletter.lms.ac.uk, November, 2015)"The editor has provided double service: he offers English-speakers access to Baumgart"s account and provides a summary of what has happened since then. The result is a very useful book." (Fernando Q. Gouvźa, MAA Reviews, June, 2015)

Franz Lemmermeyer hat nach seiner Promotion in Heidelberg und seiner Habilitation in Bonn an Universitäten in den USA und in der Türkei gelehrt, und unterrichtet seit 2007 Mathematik am Gymnasium St. Gertrudis in Ellwangen.

Recenzijas

Baumgart collected and analyzed existing proofs of QRL in his 1885 thesis, translated here into English for the first time. Summing Up: Recommended. (D. V. Feldman, Choice, Vol. 53 (5), January, 2016)

The book has an excellent comparative discussion of many proofs along with historic notes and comments by translator. It contains a vast list of references that are updated. This excellent book is a necessary one for any number theorist. Every student in the field can find a lot of virgin ideas for further research as well. This book should be a good resource for mathematics historian as well. (Manouchehr Misaghian, zbMATH 1338.11003, 2016)

The book under review provides an English translation by Franz Lemmermeyer, who is an expert in both the history of mathematics and also in algebraic number theory, of this highly remarkable thesis. In particular, the many valuable comments of the translator make the reading a pleasure and accessible to mathematicians not trained in studying the older literature. (Jörn Steuding, London Mathematical Society Newsletter, newsletter.lms.ac.uk, November, 2015)

The editor has provided double service: he offers English-speakers access to Baumgarts account and provides a summary of what has happened since then. The result is a very useful book. (Fernando Q. Gouvźa, MAA Reviews, June, 2015)

Part I Presentation of the Proofs of the Quadratic Reciprocity Law
1 From Fermat to Legendre
3(4)
2 Gauss's Proof by Mathematical Induction
7(8)
3 Proof by Reduction
15(26)
1 Gauss's Third Proof [ 26]
15(2)
2 Gauss's Fifth Proof [ 28]
17(2)
3 Eisenstein's Geometric Proof [ 17]
19(1)
4 Proof by Genocchi [ 37]
20(2)
5 Proof by Stern [ 70]
22(2)
6 Proof by Zeller [ 73]
24(2)
7 Proof by Kronecker [ 43]
26(2)
8 Proof by Bouniakowski [ 3]
28(3)
9 Proof by Schering [ 67, 68]
31(2)
10 Proof by Petersen [ 63, 64]
33(1)
11 Proof by Voigt [ 71]
34(1)
12 Proof by Busche [ 4]
35(6)
4 Eisenstein's Proof Using Complex Analysis
41(4)
5 Proofs Using Results from Cyclotomy
45(18)
1 Proof by Gauss (7th Proof) [ 34] Gauss and Lebesgue (2nd Proof) [ 52]
45(3)
2 Gauss' Fourth Proof [ 27]
48(4)
3 Gauss' Sixth Proof [ 29]
52(3)
4 Proof by Cauchy [ 8], Cauchy Jacobi [ 56, p. 391], Jacobi Eisenstein [ 15]
55(1)
5 Second Proof by Eisenstein [ 14]
56(4)
6 Proof by Liouville [ 61]
60(1)
7 First Proof by Lebesgue [ 50]
60(3)
6 Proofs Based on the Theory of Quadratic Forms
63(8)
1 Preliminary Remarks
63(8)
1.1 Gauss's Second Proof [ 25, art. 257], [ 12, Suppl. IV, X]
64(1)
1.2 Kummer's First Proof [ 48]
65(3)
1.3 Kummer's Second Proof [ 48]
68(3)
7 The Supplementary Laws of the Quadratic Reciprocity Law and the Generalized Reciprocity Law
71(6)
1 The Supplementary Laws
71(3)
1.1 Proof of Formula (I) Using "Associate Residues" (See [ 21, p. 135], [ 25, Art. 109]), for Formula (II) Using Induction
71(1)
1.2 Proof of the Supplementary Laws Using Reduction
72(1)
1.3 Proof of the Second Supplementary Law Using Cyclotomy
73(1)
1.4 Proof of the Supplementary Laws Using Quadratic Forms [ 25, art. 262]
73(1)
2 The Generalized Reciprocity Law
74(3)
8 Algorithms for Determining the Quadratic Character
77(8)
1 Gauss's Method for Determining (a/b) [ 32, p. 59]
77(2)
2 The Algorithms by Eisenstein [ 13] and Lebesgue [ 51]
79(2)
2.1 Eisenstein
79(1)
2.2 The Algorithms of Lebesgue
79(2)
3 The Algorithms by Gegenbauer [ 35]
81(1)
3.1 The Second Method
81(1)
4 An Algorithm by Kronecker
82(3)
Part II Comparative Presentation of the Principles on Which the Proofs of the Quadratic Reciprocity Law Are Based
9 Gauss's Proof by Induction
85(4)
10 Proofs by Reduction
89(18)
11 Eisenstein's Proofs Using Complex Analysis
107(4)
12 Proofs Using Results from Cyclotomy
111(14)
13 Proofs Based on the Theory of Quadratic Forms
125(2)
14 Final Comments
127(4)
15 Proofs of the Quadratic Reciprocity Law
131(32)
Bibliography 163(4)
Author Index 167(4)
Subject Index 171
Franz Lemmermeyer received his Ph.D. from Heidelberg University and has worked at Universities in California and Turkey. He is now teaching mathematics at the Gymnasium St. Gertrudis in Ellwangen, Germany.
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