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  "body": "\n## Summary\n\nThe Hilbert symbol $(a, b)_K$ of two nonzero elements of a [[local field|J85FZEQG38]] $K$ of characteristic other than $2$ is $1$ or $-1$.\nIt is $1$ exactly when the equation $z^2 = a x^2 + b y^2$ has a solution other than $(0, 0, 0)$.\nIt is symmetric and bimultiplicative, and it is a basic tool for [[quadratic forms|0RBQS1JJCR]] over local fields.\nOver $\\Q$, the product of all local Hilbert symbols is $1$, a statement equivalent to [[quadratic reciprocity|085MJS6907]].\n\n## Explanations\n\n### L1 human\n\nThe Hilbert symbol answers a yes-or-no question: does a certain quadratic equation have a nonzero solution in the field?\nOver the real numbers, $z^2 = a x^2 + b y^2$ has a nonzero solution unless both $a$ and $b$ are negative.\nSo $(a, b)_\\R = -1$ exactly when $a \u003c 0$ and $b \u003c 0$.\nOver $\\Q_p$ the answer depends on [[Legendre symbols|0JTC045NNZ]] modulo $p$, and it can be computed by a short formula.\n",
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