Cubic Reciprocity - History

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.

Gauss's published works mention cubic residues and reciprocity three times: there is one result pertaining to cubic residues in the Disquisitiones Arithmeticae (1801). In the introduction to the fifth and sixth proofs of quadratic reciprocity (1818) 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.

From his diary and other unpublished sources, it appears that Gauss knew the rules for the cubic and quartic residuacity of integers by 1805, and discovered the full-blown theorems and proofs of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. Proofs of these were found in his posthumous papers, but it is not clear if they are his or Eisenstein's.

Jacobi published several theorems about cubic residuacity in 1827, but no proofs. In his Königsberg lectures of 1836–37 Jacobi presented proofs. The first published proofs were by Eisenstein (1844).

Read more about this topic:  Cubic Reciprocity

Famous quotes containing the word history:

    All objects, all phases of culture are alive. They have voices. They speak of their history and interrelatedness. And they are all talking at once!
    Camille Paglia (b. 1947)

    No event in American history is more misunderstood than the Vietnam War. It was misreported then, and it is misremembered now.
    Richard M. Nixon (b. 1913)

    False history gets made all day, any day,
    the truth of the new is never on the news
    False history gets written every day
    ...
    the lesbian archaeologist watches herself
    sifting her own life out from the shards she’s piecing,
    asking the clay all questions but her own.
    Adrienne Rich (b. 1929)