Let $A$ be an integral domain, let $K$ be its field of fractions, and let $L$ be a finite extension of $K$. For a $\alpha \in L$, let $B=A[\alpha]$. Prove that exists a non-zero $a\in A$ such that the extension $A_a \subseteq B_a$ is integral.
Note: $A_a = S^{-1}A = \{m/n : m\in A, n\in S \}$, $S = \{1,a, a^2, a^3,...\}$. (Same for $B_a$)
Hint: Take the minimal polynomial of $\alpha$. Consider the product of all denominators in the coefficients. This will give you a localization where $\alpha$ is integral, and that's sufficient.