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  "body": "\n## Statement\n\nLet $\\Z[i]$ be the ring of [[Gaussian integers|074P7C85XH]], with norm $N(a + bi) = a^2 + b^2$.\nLet $\\pi$ be a prime element of $\\Z[i]$ that does not divide $1 + i$.\nFor $\\alpha \\in \\Z[i]$ not divisible by $\\pi$, the quartic residue character $\\left(\\frac{\\alpha}{\\pi}\\right)_4$ is the unique element of $\\set{1, i, -1, -i}$ with\n$$\\alpha^{(N(\\pi) - 1)/4} \\equiv \\left(\\frac{\\alpha}{\\pi}\\right)_4 \\pmod{\\pi}.$$\nA prime $\\pi$ is called primary if $\\pi \\equiv 1 \\pmod{(1 + i)^3}$.\nLet $\\lambda$ and $\\pi$ be primary primes of $\\Z[i]$ that are not associates.\nThen\n$$\\left(\\frac{\\lambda}{\\pi}\\right)_4 = \\left(\\frac{\\pi}{\\lambda}\\right)_4 \\cdot (-1)^{\\frac{N(\\lambda) - 1}{4} \\cdot \\frac{N(\\pi) - 1}{4}}.$$\n\n## Remarks\n\nThe proof follows the same pattern as for cubic reciprocity, with quartic [[Gauss sums|EQ6A3ZN48K]] and [[Jacobi sums|2767J23XHN]] over the field with $p$ elements for $p \\equiv 1 \\pmod 4$.\nEach prime of $\\Z[i]$ not dividing $1 + i$ has exactly one primary associate.\nThe sign is $-1$ only when both $\\lambda$ and $\\pi$ are congruent to $3 + 2i$ modulo $4$.\nA classical consequence: for a prime $p \\equiv 1 \\pmod 8$, the integer $2$ is a fourth power modulo $p$ if and only if $p = x^2 + 64y^2$ for some integers $x, y$.\nFor example, $73 = 3^2 + 64 \\cdot 1^2$, and $2^{18} \\equiv 1 \\pmod{73}$, so $2$ is a fourth power modulo $73$.\nThe prime $17$ is not of this form, and the nonzero fourth powers modulo $17$ are $1$, $4$, $13$ and $16$.\n\n## Explanations\n\n### L1 human\n\nQuartic reciprocity is the analogue of [[quadratic reciprocity|085MJS6907]] for fourth powers.\nIt compares \"is $\\lambda$ a fourth power modulo $\\pi$?\" with \"is $\\pi$ a fourth power modulo $\\lambda$?\".\nThe natural place to state it is the Gaussian integers, which contain the fourth roots of unity $\\pm 1, \\pm i$.\nThe two answers agree up to an explicit sign.\n\n### L2 human\n\nWith supplementary laws for $i$ and $1 + i$, the law lets one compute any quartic residue character by swapping and reducing.\nIt answers questions such as when $2$ is a fourth power modulo a prime, which quadratic reciprocity cannot decide.\nLike cubic reciprocity, it is a special case of the [[Artin reciprocity law|7938EKMMVR]].\n",
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