1. A.M. Legendre, Recherches d’analyse indéterminée, Histoire de l’Academie Royale des Sciences de Paris (1785), 465–559, Paris 1788 Legendre Legendre gives a partial proof of the “loi de réciprocité”, which was later turned into a full proof by Kummer Kummer using analytic techniques due to Dirichlet. Dirichlet In his book “Essai de théorie des nombres”, published in variuous editions between 1798 and 1830, Legendre gave variants of this proof but was not able to fill the gaps completely. See p. 4.
2. C.F. Gauss, Disquisitiones Arithmeticae, Braunschweig 1801. 146 Untersuchungen über die Primzahlen, deren Reste oder Nichtreste gegebene Zahlen sind, art. 107 ff. By extending a technique already used by Fermat Fermat and Euler Euler , Gauss Gauss proves the quadratic reciprocity law using induction. The actual proof begins in art. 135 of the Disquisitiones. See p. 7and p. 85.
3. C.F. Gauss, Disquisitiones Arithmeticae, Braunschweig 1801. Zweiter Beweis des Fundamentalsatzes und der übrigen auf die Reste − 1, + 2, − 2 sich beziehenden Sätze, art. 262 ff. Using his theory of binary quadratic quadratic forms binary forms, in particular the theory of ambiguous forms, Gauss gives a second proof of the quadratic reciprocity law. The technique involved was extended by Kummer Kummer in his proof of the ℓ-th power reciprocity law in cyclotomic number fields, and was subsequently generalized into a substantial piece of class class field theory field theory. See p. 63.
4. C.F. Gauss, Theorematis arithmetici demonstratio nova, Comment. Soc. regiae sci. Göttingen XVI (1808), 69
5. Werke II (1863), p. 1-8 140 Gauss 's third proof using Gauss 's Lemma became extremely popular, and was used for many (if not the most) proofs published afterwards. See p. 15 and p. 90.