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  "body": "\n## Description\n\nFor each prime power $q = p^n$, the Galois field $\\mathrm{GF}(q)$, also written $\\mathbb{F}_q$, is the [[finite field|08RTDHX1SK]] with $q$ elements.\nIt [[exists|0T8M3YC5RG]], and it is [[unique up to isomorphism|010PKCQPXP]].\nOne model is the splitting field of $x^q - x$ over $\\mathbb{F}_p = \\mathbb{Z}/p\\mathbb{Z}$.\nAnother model is $\\mathbb{F}_p[x]/(f)$ for an irreducible polynomial $f \\in \\mathbb{F}_p[x]$ of degree $n$.\n\n## Remarks\n\nThe name honours Évariste Galois.\nThe field $\\mathrm{GF}(4) = \\mathbb{F}_2[x]/(x^2 + x + 1)$ has the elements $0, 1, \\alpha, \\alpha + 1$, with $\\alpha^2 = \\alpha + 1$.\nThe field $\\mathrm{GF}(p^m)$ is isomorphic to a subfield of $\\mathrm{GF}(p^n)$ if and only if $m$ divides $n$.\n\n## Explanations\n\n### L1 human\n\nSince there is only one finite field of each order, up to a change of names, one gives it a fixed name: the Galois field with $q$ elements.\nComputations in it are done with polynomials over the integers modulo $p$.\n",
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