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  "body": "\n## Definition\n\nLet $p$ be a prime.\nOrder the [[monic|0BDX4MBKT4]] polynomials of degree $n$ over $\\F_p$ as follows: write such a polynomial as\n$$x^n - a_{n-1} x^{n-1} + a_{n-2} x^{n-2} - \\cdots + (-1)^n a_0$$\nwith $a_i \\in \\set{0, 1, \\dots, p - 1}$, and compare the sequences $(a_{n-1}, a_{n-2}, \\dots, a_0)$ lexicographically.\nThe *Conway polynomial* $C_{p,n}$ is defined by induction on $n \\ge 1$.\nIt is the least monic polynomial $f$ of degree $n$ over $\\F_p$ for this order such that:\n\n1. $f$ is a [[primitive polynomial|HQ7KMZYPSQ]] over $\\F_p$;\n2. for every divisor $m$ of $n$ with $m \u003c n$, the polynomial $f(x)$ divides $C_{p,m}\\left(x^{(p^n - 1)/(p^m - 1)}\\right)$.\n\n## Remarks\n\nClause 2 says: if $\\alpha$ is a root of $C_{p,n}$, then $\\alpha^{(p^n - 1)/(p^m - 1)}$ is a root of $C_{p,m}$.\nSo the copies of $\\F_{p^m}$ inside the models of $\\F_{p^n}$ match for all $m$ dividing $n$.\nExample: $C_{2,3} = x^3 + x + 1$.\nThe primitive cubics over $\\F_2$ are $x^3 + x + 1$ and $x^3 + x^2 + 1$, with sequences $(0, 1, 1)$ and $(1, 0, 1)$.\nClause 2 holds for both, since $C_{2,1} = x + 1$ has root $1$ and every power of $1$ is $1$.\nNon-example: $x^4 + x^3 + x^2 + x + 1$ is irreducible over $\\F_2$ but fails clause 1, so it is not $C_{2,4}$, which is $x^4 + x + 1$.\nFor $p = 5$, $C_{5,1} = x - 2 = x + 3$, since $1$ is not a primitive root modulo $5$ and $2$ is.\n",
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    "name": "Conway polynomial",
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