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  "body": "\n## Statement\n\nLet $F$ be a [[finite field|08RTDHX1SK]] of order $q = p^n$, with $p$ prime.\nThen $F$ is a [[splitting field|02NZZ1J0FH]] of $x^q - x$ over its prime subfield $\\mathbb{F}_p$.\n\n## Remarks\n\nThe proof: [[$x^q - x = \\prod_{a \\in F} (x - a)$|0QWR8WTZEW]], so $x^q - x$ splits over $F$, and its roots are all the elements of $F$. So the roots generate $F$.\nWith the [[uniqueness of splitting fields|069A3BC3EP]], this gives the [[uniqueness of finite fields|010PKCQPXP]].\n\n## Explanations\n\n### L1 human\n\nA finite field consists exactly of the roots of $x^q - x$.\nSo it is the smallest field that contains all these roots.\n",
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