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  "body": "\n## Definition\n\nLet $K$ be a [[local field|J85FZEQG38]] whose characteristic is not $2$.\nFor $a, b \\in K^\\times$, the *Hilbert symbol* $(a, b)_K$ is\n$$(a, b)_K = \\begin{cases} 1 \u0026 \\text{if } z^2 = a x^2 + b y^2 \\text{ has a solution } (x, y, z) \\in K^3 \\setminus \\set{(0, 0, 0)}, \\\\ -1 \u0026 \\text{otherwise.} \\end{cases}$$\n\n## Remarks\n\nThe value only depends on the classes of $a$ and $b$ modulo squares.\nEquivalently, $(a, b)_K = 1$ exactly when $a$ is a norm from the field $K(\\sqrt{b})$.\nThe symbol is symmetric and bimultiplicative, and $(a, 1 - a)_K = 1$ for $a \\neq 1$.\nExample: $(-1, -1)_\\R = -1$, because $z^2 = -x^2 - y^2$ forces $x = y = z = 0$ over $\\R$.\nExample: $(2, 7)_\\R = 1$, because $(x, y, z) = (1, 1, 3)$ solves $z^2 = 2x^2 + 7y^2$.\nNon-example: the solution $(0, 0, 0)$ must be excluded; it solves every such equation, so allowing it would make every symbol equal to $1$.\nFor $K = \\Q_p$ with $p$ odd, write $a = p^\\alpha u$ and $b = p^\\beta v$ with units $u, v \\in \\Z_p^\\times$.\nThen $(a, b)_p = (-1)^{\\alpha \\beta (p - 1)/2} \\left(\\frac{u}{p}\\right)^\\beta \\left(\\frac{v}{p}\\right)^\\alpha$, with [[Legendre symbols|0JTC045NNZ]] of the residues of $u$ and $v$.\n",
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