Let $A =\mathbb{Z[x]}$ be the ring of polynomials of one variable over $\mathbb{Z}$. Let $K$ be its field of fractions, $L/K$ be a finite extension, $B$ the integral closure of $A$. Since $A$ is a Noetherian integrally closed domain and $L/K$ is a separable extension, $B$ is also a Noetherian integrally closed domain.
Let $P$ be a prime ideal of height 1 of $B$. Let $p = A\cap P$. We say $P$ is inseparable(over $p$) if the field of fractions of $B/P$ is inseparable over the field of fractions of $A/p$.
Are there inseparable prime ideals of height 1 in $B$?
The motivation came from this question: https://math.stackexchange.com/questions/1190280/ramification-theory-on-noetherian-integrally-closed-domains