This is from Algebraic Number Theory by Neukirch
Let $A$ be an integral
domain which is integrally closed, K its field of fractions, $L|K$ a finite
field extension, and $B$ the integral closure of $A$ in $L$.
Furthermore, the fact that $A$ is integrally closed has the effect that an element $\beta \in L$ is integral over $A$ if and only if its minimal polynomial $p(x)$ takes its coefficients in $A$. In fact, let $\beta$ be a zero of the monic polynomial $g(x) \in A[x]$.Then $p(x)$ divides $g(x)$ in $K[x]$, so that all zeroes $\beta_1, …, \beta_n$ of $p(x)$ are integral over A, hence the same holds for all the coefficients, in other
words $p(x) \in A[x]$.
Now my question is are $\beta_1, …, \beta_n$ all the roots of $p(x)$ in $L$ or $\bar K$, the algebraic closure of $K$? I think this arguement will only work when we consider all the roots of $p(x)$ in $\bar K$.