Let $K$ be a non-archimedean local field. Then the local existence theorem states that the norm groups in $K^*$ are exactly the open subgroups of finite index.
Here what is meant by an 'open' subgroup of $K^*$? I am getting confused, and I would appreciate any clarification. Thank you.
Best Answer
Let's consider the easy example where $K = \Bbb Q_p$. It has a topology, induced by the $p$-adic metric, namely $d(x,y) = |x-y|_p$, where $| \cdot |_p$ is the $p$-adic absolute value.
Then the subspace $K^{\times} \subset K$ has an induced topology, and a subset $U \subset K^{\times}$ is open iff it is the intersection of $K^{\times}$ with a union of open balls $B_{d}(x,r)$ for some $x \in K^{\times}, r>0$.
Then a subgroup $H \leq K^{\times}$ is open (with respect to the $p$-adic topology) of finite index iff it is a norm subgroup.