Let $A \subset B$ be a ring extension, and let $f,g \in B[x]$ be monic polynomials such that $fg \in A[x]$. Prove that the coefficients of $f$ and $g$ are integral over $A$.
My attempt was to prove that $A[y]$ is finite (as an $A$-module), for every coefficients $y$ from either $f$ or $g$, but I cant make it. Im not looking for solution but for hints.
Consider a ring $R$ containing $B$ where $f$ and $g$ split into linear factors. (For monic polynomials such a ring there always exists.) Then show that the root of $f$ and $g$ in $R$ are integral over $A$.