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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.


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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
thanks. i think about this – Camilo Acevedo. Feb 15 '13 at 0:12
up vote 1 down vote accepted

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.

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thanks people, I think the same. – Camilo Acevedo. Feb 14 '13 at 4:48

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