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Cubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence ''x''3 ≡ ''p'' (mod ''q'') is solvable; the word "reciprocity" comes from the form of the main theorem, which states that if ''p'' and ''q'' are primary numbers in the ring of Eisenstein integers, both coprime to 3, the congruence ''x''3 ≡ ''p'' (mod ''q'') is solvable if and only if ''x''3 ≡ ''q'' (mod ''p'') is solvable. ==History== Sometime before 1748 Euler made the first conjectures about the cubic residuacity of small integers, but they were not published until 1849, after his death.〔Euler, ''Tractatus ...'', §§ 407–410〕 Gauss's published works mention cubic residues and reciprocity three times: there is one result pertaining to cubic residues in the Disquisitiones Arithmeticae (1801).〔Gauss, DA, footnote to art. 358〕 In the introduction to the fifth and sixth proofs of quadratic reciprocity (1818)〔Gauss, ''Theorematis fundamentalis ...''〕 he said that he was publishing these proofs because their techniques (Gauss's lemma and Gaussian sums, respectively) can be applied to cubic and biquadratic reciprocity. Finally, a footnote in the second (of two) monographs on biquadratic reciprocity (1832) states that cubic reciprocity is most easily described in the ring of Eisenstein integers.〔Gauss, BQ, § 30〕 From his diary and other unpublished sources, it appears that Gauss knew the rules for the cubic and quartic residuacity of integers by 1805, and discovered the full-blown theorems and proofs of cubic and biquadratic reciprocity around 1814.〔Cox, pp. 83–90〕〔Lemmermeyer, pp. 199–201, 222–224〕 Proofs of these were found in his posthumous papers, but it is not clear if they are his or Eisenstein's.〔Lemmermeyer, p. 200〕 Jacobi published several theorems about cubic residuacity in 1827, but no proofs.〔Jacobi, ''De residuis cubicis ...''.〕 In his Königsberg lectures of 1836–37 Jacobi presented proofs.〔 The first published proofs were by Eisenstein (1844).〔Eisenstein, ''Beweis des Reciprocitätssatzes ...''〕〔Eisenstein, ''Nachtrag zum cubischen...''〕〔Eisenstein, ''Application de l'algèbre...''〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Cubic reciprocity」の詳細全文を読む スポンサード リンク
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