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)
