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  "body": "\n## Statement\n\nLet $p$ be a prime number and let $n \\ge 1$.\nThen there is a [[finite field|08RTDHX1SK]] of order $q = p^n$.\n\n## Remarks\n\nThe proof: let $L$ be a [[splitting field|01441KDGBY]] of $f = x^q - x$ over $\\mathbb{F}_p$. The [[derivative|0AD4VS02H3]] $f' = q x^{q-1} - 1 = -1$ has no common root with $f$, so $f$ has $q$ distinct roots in $L$, by the [[criterion for repeated roots|067F8YVYMH]].\nThe roots of $f$ form a subfield of $L$, because $(a + b)^q = a^q + b^q$ and $(a b)^q = a^q b^q$. This subfield has $q$ elements.\nSo for each $n$ there is an irreducible polynomial of degree $n$ over $\\mathbb{F}_p$.\n\n## History\n\nE. H. Moore first proved the classification of finite fields in 1893: there is a field of each prime power order, and all fields of a given order are isomorphic.\n\n## Explanations\n\n### L1 human\n\nThe field with $p^n$ elements is made of the roots of $x^{p^n} - x$.\nThe proof checks that these roots are distinct and closed under the field operations.\n",
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