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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.

Jack
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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 $A$.

user26857
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  • Why does such a ring always exist for monic polynomials? –  Oct 26 '14 at 23:49
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    @Sodan Try to prove this by induction on the degree of the polynomial. In fact, the proof is similar to the one in field theory that shows that any polynomial has a splitting field. – user26857 Oct 26 '14 at 23:52