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  "body": "\n## Statement\n\nLet $\\omega = e^{2\\pi i/3}$ be a primitive cube [[root of unity|0YV2CKP2AV]], and let $\\Z[\\omega] = \\set{a + b\\omega : a, b \\in \\Z}$ be the ring of Eisenstein integers, with norm $N(a + b\\omega) = a^2 - ab + b^2$.\nLet $\\pi$ be a prime element of $\\Z[\\omega]$ with $N(\\pi) \\ne 3$.\nFor $\\alpha \\in \\Z[\\omega]$ not divisible by $\\pi$, the cubic residue character $\\left(\\frac{\\alpha}{\\pi}\\right)_3$ is the unique element of $\\set{1, \\omega, \\omega^2}$ with\n$$\\alpha^{(N(\\pi) - 1)/3} \\equiv \\left(\\frac{\\alpha}{\\pi}\\right)_3 \\pmod{\\pi}.$$\nA prime $\\pi$ is called primary if $\\pi \\equiv 2 \\pmod 3$ in $\\Z[\\omega]$.\nLet $\\pi_1$ and $\\pi_2$ be primary primes of $\\Z[\\omega]$ with $N(\\pi_1) \\ne 3$, $N(\\pi_2) \\ne 3$ and $N(\\pi_1) \\ne N(\\pi_2)$.\nThen\n$$\\left(\\frac{\\pi_1}{\\pi_2}\\right)_3 = \\left(\\frac{\\pi_2}{\\pi_1}\\right)_3.$$\n\n## Remarks\n\nThe standard proof uses cubic [[Gauss sums|EQ6A3ZN48K]] and [[Jacobi sums|2767J23XHN]].\nFor a prime $\\pi$ of norm $p \\equiv 1 \\pmod 3$, the Jacobi sum of the cubic character is a primary element of norm $p$.\nThe cube of the Gauss sum equals $p$ times this Jacobi sum.\nComparing this identity modulo the other prime gives the law.\nEach prime of $\\Z[\\omega]$ not dividing $3$ has exactly one primary associate, so the hypothesis \"primary\" only fixes the choice among the six associates.\nA classical consequence: for a prime $p \\equiv 1 \\pmod 3$, the integer $2$ is a cube modulo $p$ if and only if $p = x^2 + 27y^2$ for some integers $x, y$.\nFor example, $31 = 2^2 + 27 \\cdot 1^2$, and indeed $4^3 = 64 \\equiv 2 \\pmod{31}$.\nThe prime $7$ is not of this form, and the cubes modulo $7$ are only $0$, $1$ and $6$.\n\n## Explanations\n\n### L1 human\n\nCubic reciprocity is the analogue of [[quadratic reciprocity|085MJS6907]] for cubes instead of squares.\nIt relates the question \"is $\\pi_1$ a cube modulo $\\pi_2$?\" to the question \"is $\\pi_2$ a cube modulo $\\pi_1$?\".\nTo state it cleanly one must work in the Eisenstein integers, which contain the cube roots of unity.\nThere the answer to both questions is the same.\n\n### L2 human\n\nThe law is usually completed by supplementary laws for the units and for the prime $1 - \\omega$ above $3$.\nTogether they let one compute any cubic residue character by repeated swapping and reduction, just as with the Legendre symbol.\nThe proof through Gauss and Jacobi sums is the model for the higher reciprocity laws, and the general case is part of class field theory through the [[Artin reciprocity law|7938EKMMVR]].\n",
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