Can a generator of the ring of integers of local fields can be chosen so that it is also a uniformizer at the same time? Let $L/K$ be an extension of local fields. We can find $\alpha$ such that $\mathcal{O}_L=\mathcal{O}_K[\alpha]$. What do we know about this generating element? I think that this $\alpha$ can be selected in such a way that, in addition to the above property, it is also a uniformizing parameter at the same time. But I can not prove it.

Hint: If $\mathcal{O}_L = \mathcal{O}_K[\alpha]$ with $\alpha$ a uniformiser of $\mathcal{O}_L$, then the canonical inclusion of their residue fields $\mathcal{O}_K/(\pi_K) \hookrightarrow \mathcal{O}_L/(\alpha)$  is surjective.
Conversely, if the extension is totally ramified (which is what the above says), then ...
reuns' answer says the same and more.
