Let $A$ be a subring of $B$ and $C$ is the integral closure of the ring $A$ in $B$. Then show that $C[x]$ is the integral closure of $A[x]$ in $B[x]$.
Can anyone help me out with this? One direction is seemed to be easier. If $f\in C[x]$ is of the form $f=\sum_{i=0}^n{c_ix^i}$. Then $A[c_0,c_1,\dots,c_n]$ is a finitely-generated $A$-module, since $c_0,c_1,\dots,c_n$ are integral over $A$. Can we now conclude from here that $A[x][c_0,c_1,\dots,c_n]=A[x,c_0,c_1,\dots,c_n]$ is a finitely generated $A[x]$ module and hence $f$ is integral over A[x]?
