If the product of two rational polynomials is an integer polynomial, then the pairwise product of any two coefficients is an integer

Let f(x), g(x) \in \mathbb{Q}[x] such that f(x)g(x) \in \mathbb{Z}[x]. Prove that the product of any coefficient of f(x) with any coefficient of g(x) is an integer.

Note that f(x)g(x) is in \mathbb{Z}[x] and factors in \mathbb{Q}[x]. By Gauss’ Lemma, there exist r,s \in \mathbb{Q} such that rf, sg \in \mathbb{Z}[x] and (rf)(sg) = fg. Since \mathbb{Q} is an integral domain, in fact rs = 1. Let f_i and g_i denote the coefficients of f and g, respectively; we have rf_i \in \mathbb{Z} and r^{-1}g_i \in \mathbb{Z}, so that f_ig_j \in \mathbb{Z} for all i and j.

Note that the proof still works if we replace \mathbb{Z} by an arbitrary unique factorization domain and \mathbb{Q} by its field of fractions.

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  • Will  On December 9, 2011 at 5:36 am

    Sorry if this is a silly question, but could you explain why Q[x] being an integral domain forces rs = 1?

    • nbloomf  On December 9, 2011 at 10:44 am

      Look at the leading coefficient of (rf)(sg) = fg. (Perhaps a better justification would be ‘since \mathbb{Q} is an integral domain’.)

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