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  "body": "\n## Definition\n\nLet $F$ be a [[finite field|08RTDHX1SK]] of order $q$.\nAn element $\\alpha \\in F$ is a primitive element of $F$ if $\\alpha$ generates the multiplicative group $F^\\times$.\nThat is, $F^\\times = \\{ \\alpha, \\alpha^2, \\dots, \\alpha^{q-1} = 1 \\}$.\n\n## Remarks\n\nPrimitive elements exist, because [[$F^\\times$ is cyclic|0EBZ3G455G]]. There are exactly $\\varphi(q - 1)$ of them, where $\\varphi$ is Euler's totient function.\nThe element $2$ is a primitive element of $\\mathbb{F}_3$ and of $\\mathbb{F}_5$, but not of $\\mathbb{F}_7$. The element $3$ is a primitive element of $\\mathbb{F}_7$.\nA primitive element $\\alpha$ of $F$ generates $F$ as a [[simple extension|0P6SBR7YM2]] of its prime subfield. The converse is false in general.\n\n## Explanations\n\n### L1 human\n\nA primitive element is one whose powers give every nonzero element of the field.\nIt works like a base for logarithms in the finite field.\n",
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