Is the trace map open for $p$-adic field?

Question

Let $K$ be a finite extension of $\mathbb{Q}_p$, and $L$ be a finite extension of $K$, then is the trace map $O_L\rightarrow O_K$ open? I guess this should be true, but I don't know how to prove.

Thanks!

Answer 1

If $p\nmid [L:K]$ then $$Tr_{L/K}(\pi_K^n O_L) = \pi_K^nO_K$$ Do you think the case $p\ |\ [K:\Bbb{Q}_p]$ changes something ? (try with $K=\Bbb{Q}_2(2^{1/2})$)