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  "body": "\n## Statement\n\nLet $p$ be an odd prime, and let $a$ and $b$ be integers.\nThen\n$$\\left(\\frac{ab}{p}\\right) = \\left(\\frac{a}{p}\\right) \\left(\\frac{b}{p}\\right),$$\nwhere the symbols are [[Legendre symbols|01J8AMV9RS]].\n\n## Remarks\n\nThe proof follows from [[Euler's criterion|0M7DEDKSYB]], because $(ab)^{(p-1)/2} = a^{(p-1)/2} b^{(p-1)/2}$.\nIn words: modulo $p$, the product of two nonresidues is a residue, and the product of a residue and a nonresidue is a nonresidue.\nFor example, $3$ and $5$ are nonresidues modulo $7$, and $15 \\equiv 1 \\pmod 7$ is a residue.\nSo the map $a \\mapsto \\left(\\frac{a}{p}\\right)$ is a completely multiplicative function of $a$.\n\n## Explanations\n\n### L1 human\n\nTo decide whether a product is a square modulo $p$, you can test each factor.\nTwo \"no\" answers give \"yes\", just as two negative signs give a positive sign.\n",
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  "msc": [
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  "name": "Multiplicativity of the Legendre symbol",
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