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.

$O_L=O_K[\pi_L]$ iff $f(L/K)=1$ and $O_L=O_K[\zeta_{q_L-1}]$ iff $e(L/K)=1$. In general it is $$O_L=O_K[\zeta_{q_L-1}+\pi_L]$$
Hensel lemma is needed to construct the root unity, then (in the non-trivial case $e\ne 1,f\ne 1$) we need a closed-ness and density argument, investigating $\zeta_{q_L-1}+\pi_L-(\zeta_{q_L-1}+\pi_L)^q$
