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  "body": "\n## Definition\n\nLet $\\F_q$ be a [[finite field|08RTDHX1SK]] and let $m \\ge 1$.\nA polynomial $f \\in \\F_q[x]$ of degree $m$ is *primitive* if it is [[irreducible|00X2A9G9W3]] and has a root $\\alpha$ in $\\F_{q^m}$ that is a [[primitive element|0YAA9CJQXF]].\nThat is, $\\alpha$ has multiplicative [[order|0JPYSD5E0B]] $q^m - 1$.\n\n## Remarks\n\nEquivalently, $f$ is a scalar multiple of the [[minimal polynomial|0GK1WB9R98]] over $\\F_q$ of a primitive element of $\\F_{q^m}$.\nThen all roots of $f$ are primitive elements, since they are the conjugates $\\alpha^{q^i}$, which have the same order as $\\alpha$.\nAnother standard form: an irreducible $f$ of degree $m$ with $f(0) \\ne 0$ is primitive if and only if the least $e \\ge 1$ such that $f$ divides $x^e - 1$ is $q^m - 1$.\nThere are $\\varphi(q^m - 1)/m$ monic primitive polynomials of degree $m$ over $\\F_q$, where $\\varphi$ is [[Euler's totient function|0VV04M41AN]].\nExample: over $\\F_2$, the polynomial $x^3 + x + 1$ is irreducible, and a root $\\alpha$ has order dividing $7$ but not $1$, so its order is $7 = 2^3 - 1$, and the polynomial is primitive.\nNon-example: over $\\F_2$, the polynomial $x^4 + x^3 + x^2 + x + 1$ is irreducible, but it divides $x^5 - 1$, so a root has order $5$, not $15$, and the polynomial is not primitive.\n",
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