Let $R$ be a domain with field of fractions $K$ and suppose that $t$ is a transcendental element over $K$. Consider $f(t)=\sum_{k=0}^l a_k t^k,\ \in K[t]$. Show that if $f(t)$ is integral over $R[t]$, then $a_k$ is integral over $R$ for each $k$.
My approach: I have thus far only been able to prove that $a_0,a_l$ are integral over $R$ the following way:
As $f(t)$ is integral, there exists such a $g(t)\in R[t]$ s.t. $g(t)=f^m(t)+f^{m-1}(t)\cdot r_{m-1}+\dots+r_0=0$ where $r_i\in R$. Thus, if we plug in our sum for $f(t)$, we could regroup $g$ in descending order of $t^i$. As $t\neq 0$, one needs the coefficients to be equal to zero. From here, it is easy to see that $a_0,a_l$ are integral. However, not so much the rest, as I can't get a monic polynomial for them. Could someone help me?
