Gauss's Disquitiones Arithmeticae centers around the quadratic reciprocity law. It seems that he developed the genus theory of integral binary quadratic forms to find a natural proof of the quadratic reciprocity law. He later wrote 5 papers on number theory. All of them are about the quadratic and biquadratic reciprocity laws. Why did Gauss think the reciprocity law so important in number theory?

  • $\begingroup$ Quadratic forms are the primary focus, Reciprocity more a powerful tool. $\endgroup$ Aug 19 '13 at 0:12
  • $\begingroup$ @AndréNicolas Gauss proved the quadratic reciprocity using the theory of quadratic forms. Other than this, if he thought the quadratic reciprocity more of a tool, it is hard to explain why he provided 7 different proofs of the reciprocity throughout his life. $\endgroup$ Aug 19 '13 at 5:35
  • $\begingroup$ Quadratic forms had a long history by then, and reciprocity had been spotted much earlier as relevant by Euler and Legendre. $\endgroup$ Aug 19 '13 at 5:39
  • $\begingroup$ @AndréNicolas If my memory is correct, Gauss developed the theory of quadratic forms independently from other mathematicians. The same for the quadratic reciprocity. $\endgroup$ Aug 19 '13 at 5:51

It's the first non-trivial result in elementary number theory, and to find out whether a second-degree equation has a solution modulo some number, basically comes down to knowing quadratic reciprocity. The Disquisitiones starts with the basic definition of congruences, then solutions of first-degree congruence equations, and then second-degree, where one naturally bumps up against this problem.

Here is from the book itself:

Since almost everything that can be said about quadratic residues depends on this theorem, the term fundamental theorem which we will use from now on should be acceptable.

Later on he speaks of how Euler and Legendre came close to finding a proof, but despite their best efforts, could not. Clearly it was also a matter of pride for him.

Given how much of modern number theory revolves around generalizations of quadratic reciprocity, such as Artin reciprocity and the reciprocity conjecture of Langlands, it's easy to consider Gauss's intuition as justified.

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    $\begingroup$ I corrected a typo that said Lagrange instead of Legendre. Gauss specifically refers to Legendre's "Recherches d'analyse indétérminée" Hist. Acad. Paris, 1785, p.465 ff $\endgroup$
    – Zavosh
    Aug 19 '13 at 0:39
  • $\begingroup$ Thanks. Please notice that by the reciprocity law I mean not only the quadratic reciprocity law but also higher reciprocity law. Gauss clearly thought higher reciprocity law also very important. $\endgroup$ Aug 19 '13 at 1:04
  • $\begingroup$ The higher reciprocity laws were at first attempts to generalize quadratic reciprocity. Naturally, if you know how to solve quadratic congruence equations, next you want to solve cubics, hence cubic reciprocity, biquadratic, etc. The most general reciprocity law (Artin's) did not appear until a hundred years later, so it makes sense that Gauss would not be satisfied so soon. $\endgroup$
    – Zavosh
    Aug 25 '13 at 10:05
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    $\begingroup$ The fantastic book 'Reciprocity Laws: from Euler to Eisenstein' by F. Lemmermeyer has the history as well as the math in great detail. $\endgroup$
    – Zavosh
    Aug 25 '13 at 10:07

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