Dirichlet’s Lectures on Quadratic Forms

  • Jeremy Gray
Part of the Springer Undergraduate Mathematics Series book series (SUMS)


Gauss’s first proof of quadratic reciprocity was given in Section IV of the Disquisitiones Arithmeticae. Gauss went on to publish five more, leaving a further two unpublished; he also knew that these different proofs hinted at important connections to as-yet undiscovered parts of mathematics. But insofar as some of these proofs were intended to explore or illustrate these connections they were of varying levels difficulty and not all suitable for beginners. The simplest is the third proof, which was adopted by Peter Gustav Lejeune Dirichlet in lectures that he gave in the 1850s and which formed part of his book Vorlesungen über Zahlentheorie (Lectures on Number Theory) that did so much to bring number theory to a wide audience of mathematicians. We look at this proof here, and then turn to look at Dirichlet’s Lectures more broadly.


  1. Dirichlet, P.G.L.: Recherches sur diverses applications de l’analyse infinitésimale à la théorie des nombres. J. Math. 19, 324–369; 20, 1–12; 134– 155 (1839/1840); in Gesammelte Werke 1, 411–496Google Scholar
  2. Gauss, C.F.: (3rd Proof) Theorematis arithmetici demonstratio nova. Comment. Soc. regiae sci. Gottingen. In: Werke, vol. 2, pp. 1–8 (1808)Google Scholar
  3. Scharlau, W., Opolka, H.: From Fermat to Minkowski. Springer, Berlin (1984)MATHGoogle Scholar

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Authors and Affiliations

  • Jeremy Gray
    • 1
    • 2
  1. 1.School of Mathematics and StatisticsThe Open UniversityMilton KeynesUK
  2. 2.Mathematics InstituteUniversity of WarwickCoventryUK

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