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

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

• If I understand your question correctly, I'd say no, because there are no inseparable extensions of Z/p (finite fields are perfect). – John Brevik Mar 16 '15 at 4:29
• @JohnBrevik $A/p$ is not a finite field. – Makoto Kato Mar 16 '15 at 23:25
• Oops, sorry, didn't read very carefully. – John Brevik Mar 17 '15 at 0:48
• How about this, though: What if B=A[t]/(t^3-x)? Isn't 3B inseparable? – John Brevik Mar 17 '15 at 1:00
• @JohnBrevik I have at least two questions. How do you prove that $B$ is integrally closed? And how do you prove $3B$ is a prime ideal? – Makoto Kato Mar 22 '15 at 2:49

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.