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  "body": "\n## Summary\n\nOver a [[finite field|0ZXZ82FTX2]] every polynomial can be [[factored into irreducibles|JWB6B13Q8Z]] by efficient algorithms.\nThe usual methods remove repeated factors, then group the irreducible factors by degree, then split each group.\nThey rest on the fact that $x^{q^d} - x$ is the product of all monic irreducible polynomials over $\\F_q$ whose degree divides $d$.\nThese algorithms are also the first step of factoring over $\\Z$ and $\\Q$, and they are used in coding theory and cryptography.\n\n## Explanations\n\n### L1 human\n\nOver a finite field there are only finitely many polynomials of each degree, so a factorization can always be found by trying.\nGood algorithms are much faster than trying.\nThey use $\\gcd$ computations with special polynomials like $x^{q^d} - x$ to cut the polynomial into pieces.\nFor example, over $\\F_2$, $x^3 + 1 = (x + 1)(x^2 + x + 1)$, and $x^2 + x + 1$ has no root in $\\F_2$.\nThe [[definition|XXQ378NQE2]] describes the three stages.\n",
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