I stumbled upon the following problem and even though at first it seemed to be fairly easy I'm having problems with finding the proof.
We have a monic polynomial $W(x)$ with integer coefficients. We know that $W(x)=R(x)P(x)$ and that $R(x)$ is also a monic polynomial with integer coefficients. How do I prove that $P(x)$ also has all integer coefficients?
Any help is appreciated :)
Assuming "normalized" means "monic":
Suppose $W(x)=R(x)P(x)$ where $W(x)=x^n+a_{n-1}x^{n-1}+\ldots a_0$, and $R(x)=x^m+b_{m-1}x^{m-1}+\ldots+b_0$, with $a_i,b_j\in\mathbb Z$. Let $P(x)=c_\ell x^{\ell}+\ldots+c_0$ with $c_{\ell}\neq 0$, and suppose $c_{k}\notin\mathbb Z$ for some $k$.
Then in particular, there exists a largest $0\leq K\leq \ell$ such that $c_K\notin \mathbb Z$. Then for all $k\geq K$, $c_k\in \mathbb Z$. But then since $W(x)=R(x)P(x)$, $a_{m+K}=b_mc_K+z=c_K+z$, where $z=\sum_{m\geq k\geq K+1} b_{m-k}c_k\in \mathbb Z$; in particular, $c_K$ is integer. This contradicts our assumption that $c_K$ was the largest non-integer coefficient, so no such coefficient can exist; i.e. $P(x)\in\mathbb Z[x]$.