© 2015

The Quadratic Reciprocity Law

A Collection of Classical Proofs

  • Authors

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Presentation of the Proofs of the Quadratic Reciprocity Law

    1. Front Matter
      Pages 1-1
    2. Oswald Baumgart
      Pages 3-6
    3. Oswald Baumgart
      Pages 7-13
    4. Oswald Baumgart
      Pages 15-39
    5. Oswald Baumgart
      Pages 41-44
    6. Oswald Baumgart
      Pages 45-62
    7. Oswald Baumgart
      Pages 63-69
  3. Comparative Presentation of the Principles on Which the Proofs of the Quadratic Reciprocity Law Are Based

    1. Front Matter
      Pages 83-83
    2. Oswald Baumgart
      Pages 85-88
    3. Oswald Baumgart
      Pages 89-105
    4. Oswald Baumgart
      Pages 107-109
    5. Oswald Baumgart
      Pages 111-124
    6. Oswald Baumgart
      Pages 125-126
    7. Oswald Baumgart
      Pages 127-130
    8. Oswald Baumgart
      Pages 131-161
  4. Back Matter
    Pages 163-172

About this book


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.


Legendre symbols cyclotomy quadratic Gauss sums quadratic reciprocity law quadratic residues

About the authors

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.

Bibliographic information

  • Book Title The Quadratic Reciprocity Law
  • Book Subtitle A Collection of Classical Proofs
  • Authors Oswald Baumgart
  • DOI
  • Copyright Information Springer International Publishing Switzerland 2015
  • Publisher Name Birkhäuser, Cham
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Hardcover ISBN 978-3-319-16282-9
  • Softcover ISBN 978-3-319-36778-1
  • eBook ISBN 978-3-319-16283-6
  • Edition Number 1
  • Number of Pages XIV, 172
  • Number of Illustrations 1 b/w illustrations, 0 illustrations in colour
  • Topics Number Theory
  • Buy this book on publisher's site


“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,, 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. Gouvêa, MAA Reviews, June, 2015)