Let $L/K/\mathbb Q_p$ be finite extensions of local fields and let $v_L$ and $v_K$ be normalised discrete valuations on $L$ and $K$ respectively.
My question is quite a general one:
If the valuation rings are given by $\mathcal O_L=\mathcal O_K[\alpha]$ for some $\alpha\in \mathcal O_L$, then is it true that $L=K(\alpha)$ as fields?
Can this also be true in the case where $\mathcal O_K$ is a Dedekind domain, $K$ its quotient field, $L$ a finite separable extension of $K$ and $\mathcal O_L$ the integral closure of $\mathcal O_K$ in $L$?
Thank you for your help!