Let $B/A$ be a finite integral extension of discrete valuation rings of characteristic zero.
Question. Is there a uniformizer $\pi$ in $B$ such that $A[\pi]$ is a discrete valuation ring?
If not, can one give an explicit example? Note that to give such an example the extension of residue fields has to be INseparable. Is there an example with the extension of residue fields PURELY inseparable?