Inseparably ramified primes over $\mathbb{Z}[x]$

Question

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

Answer 1

OK, this is not exactly my wheelhouse, but I'm jumping in here after no answers for a few days. I've convinced myself that the following is correct, so here's hoping.

Let $p$ be a prime, and consider $B=A[t]/(t^p-x)$. $B$ is an integrally closed domain, as it is isomorphic to $\mathbb Z[x,t]/t^p-x \cong \mathbb Z[t]$ (via the surjection $t\mapsto t, x\mapsto t^p$). Also, $pB$ is a prime ideal of $B$, since $B/pB \cong (\mathbb Z/p\mathbb Z)[t]$. $pB\cap A$ is pretty clearly $pA$, and the extension in question is now $(\mathbb Z/p\mathbb Z)(x) \subseteq (\mathbb Z/p\mathbb Z)(x)[\sqrt[p]{x}]$, which is the standard example of an inseparable extension.