I define a p-adic field is a field $K$ which is a finite extension of $Q_p$ and $\pi$ is a uniformizer of $K$. And we define the norm topology on $K$ is given that the norm groups form a fundamental system of neighborhoods of 1.
Questions:
(1) Why $O_K^*$ is not open for the norm toplogy ?
(2) Let $m,n$ be two arbitary positive integers, Why $G_{m,n}:=(1+\pi^nO_K)\times \pi^m$ is a finite index open subgroup of $K^*$ for the usual metric topology on $K^*$ ? And for any finite index open subgroup $G$ in $K^*$ for the metric topology, does there exist a $G_{m,n}$ such that $G_{m,n}\subseteq G$ ?
Thanks in advance !
Each norm group, that is the nonzero norms from some finite extension $L/K$, will contain elements of nonzero valuation, so will not be contained in $O_K^*$, the set of elements of zero valuation.