Let $\mathcal O$ be a complete DVR with fraction field $K$, maximal ideal $\mathfrak p$ and residue field $\widetilde K=\mathcal O/\mathfrak p$. Now consider a subring $A\subset \mathcal O$ with the following properties:
- $A$ is a local ring with maximal ideal $\mathfrak m$ and residue field $L=A/\mathfrak m$
- $\mathcal O$ is the integral closure of $A$ in $K$.
Then, is it true or not the following claim?
$\widetilde K|L$ is a finite field extension