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  "body": "\n## Definition\n\nLet $\\F_q$ be the [[finite field|08RTDHX1SK]] with $q$ elements and let $f \\in \\F_q[x]$ be a nonzero polynomial.\nFactoring $f$ over $\\F_q$ means computing its [[factorization into irreducibles|EFDBS6ZME8]] $f = c \\, p_1^{e_1} \\cdots p_k^{e_k}$ in $\\F_q[x]$.\nThe standard algorithms do this in three stages:\n\n1. square-free factorization: write the monic part of $f$ as $g_1 g_2^2 \\cdots g_s^s$, with the $g_i$ square-free and pairwise coprime, using $\\gcd(f, f')$ for the [[formal derivative|0AD4VS02H3]] $f'$;\n2. distinct-degree factorization: for a square-free monic $g$, compute for each $d \\ge 1$ the product $h_d$ of the monic irreducible factors of $g$ of degree $d$;\n3. equal-degree factorization: split each $h_d$ into its irreducible factors of degree $d$.\n\n## Remarks\n\nStage 2 uses that $x^{q^d} - x$ is the product of all monic irreducible polynomials in $\\F_q[x]$ whose degree divides $d$.\nSo $h_1 = \\gcd(g, x^q - x)$, and $h_d$ is the [[gcd|99J1WXG3F7]] of $x^{q^d} - x$ with $g$ divided by $h_1 \\cdots h_{d-1}$.\nIn stage 1, if $f' = 0$ then $f$ is a $p$-th power, where $p$ is the characteristic, since every element of $\\F_q$ is a $p$-th power.\nStage 3 is usually done with a randomised method, such as the Cantor–Zassenhaus algorithm; [[Berlekamp's algorithm|EM8MFGFAGN]] is a deterministic alternative to stages 2 and 3.\nExample: over $\\F_2$, $g = x^3 + 1$ is square-free, $\\gcd(g, x^2 - x) = x + 1$, so $h_1 = x + 1$ and $h_2 = x^2 + x + 1$.\nNon-example: $(x + 1)(x^3 + 1) = (x + 1)^2 (x^2 + x + 1)$ is not square-free, so stage 2 does not apply before stage 1.\n\n## Explanations\n\n### L1 human\n\nFirst take out repeated factors, then sort the factors by degree, then separate factors of the same degree.\nEach stage uses only greatest common divisors and powers modulo $f$.\n",
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