Suppose that $\phi(x) = a_nx^n + \cdots + a_0x^0\in \mathbb{Z}[x]$ and let $\phi(x) = p(x)h(x)$ where $p(x),h(x)\in\mathbb{Q}[x]$ and $p(x)$ is monic. Must then $a_n\cdot p(x)$ have integer coefficents.
I am stuck on this problem. My idea is contradiction, suppose that $a_n\cdot p(x)$ doesn't have integer coefficients. Write $$ p(x) = \sum_{i=0}^{m-1}b_ix^i + x^m\\ h(x) = \sum_{j=0}^{k}c_jx^j $$ Then $a_n\phi(x) = a_np(x) h(x) = (a_nx^m+\cdots +a_nb_0)(c_kx^k+\cdots + c_0x^0)$ Looking at the leading coefficients, $$ a_n^2 = a_nc_k $$ This implies $a_n=c_n$. I don't know where to go from here.