• Winfried Scharlau
  • Hans Opolka
Part of the Undergraduate Texts in Mathematics book series (UTM)


One of the most celebrated theorems in number theory is the law of quadratic reciprocity. We formulated it at the end of Chapter 3. The history of the discovery of this theorem is complicated and not quite clear, but we will shortly show that one is led to the theorem by the problem of deciding whether a given prime number divides a number of the form x 2 + ay 2. This was how Euler and later (around 1785), independently, Legendre discovered the theorem. Unlike Euler, Lagrange tried to prove the theorem, but his proof had serious gaps. We will discuss it below. Finally, it was rediscovered by Gauss, probably after numerical calculations and not in connection with the theory of binary forms. Gauss gave the first complete proof.


Prime Number Prime Power Global Solvability Famous Theorem Binary Quadratic Form 
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  1. Y. Itard: Legendre, Adrien Marie (in Dictionary of Scientific Biography).Google Scholar
  2. A. M. Legendre: Theorie des Nombres, Paris, 1830; reprint, Blanchard, Paris 1955.Google Scholar
  3. A. M. Legendre: Zahlentheorie. (According to the Third Edition translated into German by H. Maser, Vol. II, Teubner, Leipzig, 1886.)Google Scholar
  4. C. F. Gauss: Disquisitiones Arithmeticae,Art. 151 and 296.Google Scholar
  5. Legendres correspondence with Jacobi in Jacobi, Werke,Vol. 1.Google Scholar
  6. L. Kronecker: see references to Chapter 3.Google Scholar

Copyright information

© Springer Science+Business Media New York 1985

Authors and Affiliations

  • Winfried Scharlau
    • 1
  • Hans Opolka
    • 1
  1. 1.Mathematisches InstitutUniversität MünsterMünsterWest Germany

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