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quartic reciprocity
Quartic or biquadratic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence ''x''4 ≡ ''p'' (mod ''q'') is solvable; the word "reciprocity" comes from the form of some of these theorems, in that they relate the solvability of the congruence ''x''4 ≡ ''p'' (mod ''q'') to that of ''x''4 ≡ ''q'' (mod ''p'').
==History==
Euler made the first conjectures about biquadratic reciprocity.〔Euler, ''Tractatus'', § 456〕 Gauss published two monographs on biquadratic reciprocity. In the first one (1828) he proved Euler's conjecture about the biquadratic character of 2. In the second one (1832) he stated the biquadratic reciprocity law for the Gaussian integers and proved the supplementary formulas. He said〔Gauss, BQ, § 67〕 that a third monograph would be forthcoming with the proof of the general theorem, but it never appeared. Jacobi presented proofs in his Königsberg lectures of 1836–37.〔Lemmermeyer, p. 200〕 The first published proofs were by Eisenstein.〔Eisenstein, ''Lois de reciprocite''〕〔Eisenstein, ''Einfacher Beweis ...''〕〔Eisenstein, ''Application de l'algebre ...''〕〔Eisenstein, ''Beitrage zur Theorie der elliptischen ...''〕
Since then a number of other proofs of the classical (Gaussian) version have been found,〔Lemmermeyer, pp. 199–202〕 as well as alternate statements. Lemmermeyer states that there has been an explosion of interest in the rational reciprocity laws since the 1970s.〔Lemmermeyer, p. 172〕
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