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  "body": "\n## Definition\n\nLet $K$ be a [[field|0QA8CHH8MG]] with unit element $1$.\nFor a positive integer $n$, write $n \\cdot 1 = 1 + 1 + \\dots + 1$, with $n$ terms.\nIf $n \\cdot 1 = 0$ for some positive integer $n$, the characteristic $\\operatorname{char} K$ is the least such $n$.\nOtherwise $\\operatorname{char} K = 0$.\n\n## Remarks\n\nThis is the [[characteristic of a ring|0P7CH1CND5]] applied to $K$.\nSince a field is an integral domain, the [[characteristic is zero or a prime|00758QM6JJ]].\nExample: in $\\F_5$, $5 \\cdot 1 = 0$ and $n \\cdot 1 \\ne 0$ for $1 \\le n \\le 4$, so $\\operatorname{char} \\F_5 = 5$.\nNon-example of positive characteristic: in $\\Q$, $n \\cdot 1 = n \\ne 0$ for every $n \\ge 1$, so no positive $n$ works and $\\operatorname{char} \\Q = 0$.\nThe [[prime subfield|0VPFBAY7MW]] of $K$ is isomorphic to $\\Q$ if $\\operatorname{char} K = 0$ and to $\\F_p$ if $\\operatorname{char} K = p$.\nIf $\\operatorname{char} K = p \u003e 0$, then $(a + b)^p = a^p + b^p$ for all $a, b \\in K$, which gives the [[Frobenius endomorphism|0GHDDVAH8P]].\n\n## Explanations\n\n### L1 human\n\nThe characteristic counts how many ones you must add to reach zero.\nIn the usual number fields you never reach zero, and the characteristic is $0$.\nIn a finite field you always reach zero, after a prime number of steps.\n",
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