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Suppose $R$ is a UFD, with $F$ its field of fractions.

A usual corollary to Gauss' Lemma on the content of polynomials states that if $f(X)\in F[X]$ has a factorization $f(X)=g(X)h(X)$ in $F[X]$, then if $f(X)$ and $g(X)$ are primitive and in $R[X]$, then so is $h(X)$. Why is $h(X)$ necessarily in $R[X]$?

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    You're right, my question didn't make sense. I've tried to pinpoint what I'm really asking for now.2011-10-13

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For every prime $p$ we have $0=v_p(f)=v_p(g)+v_p(h)=v_p(h)$, thus $h$ is primitive and in $R[X]$. Here, $v_p$ denotes the largest $n$ such that $p^n$ divides all the coefficients.