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  "body": "\n## Statement\n\nLet $R$ be a [[unique factorisation domain|06YMXJMXYS]] and let $f, g \\in R[x]$ be [[primitive polynomials|0SP14DZN0G]].\nThen $f g$ is primitive.\n\n## Remarks\n\nThe proof: suppose an irreducible $p$ divides all the coefficients of $f g$. Since $p$ is [[prime|0MNHXBSX3M]], $R/(p)$ is an integral domain. The images of $f$ and $g$ in $(R/(p))[x]$ are nonzero, but their product is zero, which is a contradiction.\nMany texts call this statement Gauss's lemma. The node [[Gauss's lemma|0QFV5Y1Z5X]] states the form about irreducibility.\nOver $\\mathbb{Z}$: $2x + 1$ and $3x + 2$ are primitive, and so is $6x^2 + 7x + 2$.\n\n## Explanations\n\n### L1 human\n\nSuppose that no prime divides all the coefficients of $f$, and no prime divides all the coefficients of $g$.\nThen no prime divides all the coefficients of $f g$.\nThe proof reduces the coefficients modulo a prime and uses that the result is an integral domain.\n",
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