Higher Powers
Generalizations of Gauss's lemma can be used to compute higher power residue symbols. In his second monograph on biquadratic reciprocity, Gauss used a fourth-power lemma to derive the formula for the biquadratic character of 1 + i in Z, the ring of Gaussian integers. Subsequently, Eisenstein used third- and fourth-power versions to prove cubic and quartic reciprocity.
Read more about this topic: Gauss's Lemma (number Theory)
Famous quotes containing the words higher and/or powers:
“The word which gives the key to the national vice is waste. And people who are wasteful are not wise, neither can they remain young and vigorous. In order to transmute energy to higher and more subtle levels one must first conserve it.”
—Henry Miller (1891–1980)
“The Federal Constitution has stood the test of more than a hundred years in supplying the powers that have been needed to make the Central Government as strong as it ought to be, and with this movement toward uniform legislation and agreements between the States I do not see why the Constitution may not serve our people always.”
—William Howard Taft (1857–1930)