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  "body": "\n## Definition\n\nLet $F$ be a [[field|0QA8CHH8MG]].\nThe field $F$ is *Pythagorean* if for all $a, b \\in F$ there is $c \\in F$ with $a^2 + b^2 = c^2$.\n\n## Remarks\n\nBy induction on the number of terms, every finite sum of squares in a Pythagorean field is a square.\nExample: $\\R$ is Pythagorean, since $a^2 + b^2 \\ge 0$ has the real square root $\\sqrt{a^2 + b^2}$.\nMore generally, every [[real closed field|778KFSM09P]] and every [[algebraically closed field|02SJG1HC32]] is Pythagorean.\nNon-example: $\\Q$ is not Pythagorean, because $1^2 + 1^2 = 2$ is not the square of a rational number.\nNon-example: $\\F_3$ is not Pythagorean, because $1^2 + 1^2 = 2$ and the squares in $\\F_3$ are $0$ and $1$.\n",
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