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  "body": "\n## Definition\n\nLet $K$ be a [[field|0QA8CHH8MG]] and let $f \\in K[x]$ be a nonzero polynomial.\nA factorization of $f$ into irreducibles is an expression\n$$f = c \\, p_1^{e_1} \\cdots p_k^{e_k}$$\nsuch that:\n\n1. $c \\in K$ is nonzero;\n2. $p_1, \\dots, p_k$ are pairwise distinct [[monic|0BDX4MBKT4]] [[irreducible polynomials|00X2A9G9W3]] in $K[x]$, with $k \\ge 0$;\n3. $e_1, \\dots, e_k$ are positive integers.\n\n## Remarks\n\nEvery nonzero $f$ has such a factorization, and it is unique up to the order of the factors, since $K[x]$ is a [[unique factorisation domain|06YMXJMXYS]].\nThe constant $c$ is the leading coefficient of $f$.\nExample: over $\\Q$, $2x^3 - 2x = 2 \\, x (x - 1)(x + 1)$; each factor is monic of degree $1$, hence irreducible.\nNon-example: $x^4 - 1 = (x^2 - 1)(x^2 + 1)$ is a factorization but not into irreducibles, since $x^2 - 1 = (x - 1)(x + 1)$ violates clause 2.\nOver a unique factorisation domain $R$ such as $\\Z$, a nonzero $f \\in R[x]$ is its content times a primitive polynomial.\nThe primitive part factors into primitive irreducibles, by [[polynomial ring over a UFD is a UFD|0D6KH8Y59N]].\nOver finite fields, see [[factorization over finite fields|XNE7F0BAGE]].\n\n## Explanations\n\n### L1 human\n\nThe definition fixes the shape of the answer: a constant times powers of different monic irreducible polynomials.\nThe exponents record repeated factors.\n",
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