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  "body": "\n## Definition\n\nLet $F$ be a [[finite field|08RTDHX1SK]] of characteristic $p$.\nThe Frobenius automorphism of $F$ is the map $\\varphi \\colon F \\to F$, $\\varphi(x) = x^p$.\n\n## Remarks\n\nThe map $\\varphi$ is the [[Frobenius endomorphism|0GHDDVAH8P]] of $F$. It is injective, as is every field homomorphism, so it is bijective, because $F$ is finite.\nThe map $\\varphi$ fixes every element of the prime subfield $\\mathbb{F}_p$, by Fermat's little theorem.\nIf $|F| = p^n$, then $\\varphi^n$ is the identity, and the automorphisms of $F$ are exactly $\\mathrm{id}, \\varphi, \\dots, \\varphi^{n-1}$.\nSo $\\mathrm{Gal}(\\mathbb{F}_{p^n}/\\mathbb{F}_p)$ is cyclic of order $n$, generated by $\\varphi$.\n\n## Explanations\n\n### L1 human\n\nRaising to the power $p$ respects addition in characteristic $p$, by the freshman's dream.\nIn a finite field, this map permutes the elements, and every symmetry of the field is a power of it.\n",
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