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 higher the climb, the heavier the fall.”
—Chinese proverb.
“A man is the prisoner of his power. A topical memory makes him an almanac; a talent for debate, disputant; skill to get money makes him a miser, that is, a beggar. Culture reduces these inflammations by invoking the aid of other powers against the dominant talent, and by appealing to the rank of powers. It watches success.”
—Ralph Waldo Emerson (1803–1882)