# Product of polynomials with rational coefficients

I have a homework in Algebra taken of Gallian's Contemporary Abstract Algebra pag: 303 #48.

Let $f(x)= a_nx^n + \dots +a_0$ and $g(x)= b_mx^m +\dots+b_0$ belong to $\mathbb{Q[x]}$ and suppose that $fg$ belong to $\mathbb{Z[x]}$. Prove that $a_ib_j$ is an integer for every $i$ and $j$.

Any hint will be very welcome.

Thanks.

-
Do you already know that $\mathbb{Z}$ is integrally closed? –  Martin Brandenburg Feb 13 '13 at 23:10
Do you mean $f\cdot g$? This is not true for the composition of two polynomials, consider $f(x)=x+1/2$ and $g(x)=x-1/2$. –  JSchlather Feb 14 '13 at 0:42
There is a typo in Gallian's book. There must be the product of $f$ and $g$ instead of composition. In this case you can find a proof here.