Cubic reciprocity

Cubic reciprocity is a law relating cubic residue symbols for suitable primes in the Eisenstein integers. It is the cubic analogue of quadratic reciprocity.

Cubic residue symbol: A symbol that records whether an element is a cube modulo a prime in a domain containing the cube roots of unity. The law compares these symbols for pairs of suitable Eisenstein primes.
Quadratic residue: An integer congruent to a square modulo a specified modulus. Cubic residues extend the same power-residue question from squares to cubes.
Carl Friedrich Gauss: A German mathematician whose work established major foundations of number theory and introduced several reciprocity laws. Gauss developed cubic reciprocity as part of his investigations into higher power residues.
Cubic residuacity: The property of being congruent to a cube modulo a specified prime or ideal. Cubic reciprocity transfers residuacity questions between prime moduli.
Bi-quadratic reciprocity: A reciprocity law relating fourth-power residue symbols in an appropriate cyclotomic integer ring. It raises the residue power from three to four and requires a different root-of-unity setting.
Eisenstein integers: The ring of complex numbers a + bω, where a and b are integers and ω is a primitive cube root of unity. Cubic reciprocity is formulated in this ring, where cube roots of unity are available.
Congruence: An equivalence relation stating that two integers have the same remainder modulo a fixed integer. Residue symbols encode whether congruences to cubes have solutions.
Disquisitiones Arithmeticae: Gauss's 1801 treatise that systematized number theory and proved quadratic reciprocity. Its quadratic reciprocity framework set the model for later higher reciprocity laws.
Splitting of primes in number fields: The decomposition of a rational prime into prime ideals in the ring of integers of a number field. Residue conditions in the law can be read through prime splitting in related fields.
Quadratic reciprocity law for Gaussian integers: A quadratic reciprocity theorem formulated using Gaussian integers. It shows how adjoining roots of unity can reshape reciprocity laws for another power.

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