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  "body": "\n## Statement\n\nLet $F$ be a [[finite field|08RTDHX1SK]].\nThen the [[characteristic|0P7CH1CND5]] of $F$ is a prime number $p$, and $|F| = p^n$ with $n = [F : \\mathbb{F}_p]$.\n\n## Remarks\n\nThe proof: a finite field has nonzero characteristic, and [[the characteristic of a domain is $0$ or prime|00758QM6JJ]]. The [[prime subfield|0VPFBAY7MW]] is $\\mathbb{F}_p$, and $F$ is a vector space of dimension $n$ over it, so it has $p^n$ elements.\nSo there is no field with $6$ or $10$ elements.\nThe converse, that a field of order $p^n$ exists for each $n \\ge 1$, is the [[existence theorem|0T8M3YC5RG]].\n\n## Explanations\n\n### L1 human\n\nA finite field is a vector space over the field with $p$ elements.\nA vector space of dimension $n$ over it has $p^n$ elements.\nSo the size of a finite field is always a power of a prime.\n",
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