Is any finite index subgroup of multiplicative group of p-adic field open?

Question

I found that any finite index subgroup of multiplicative group of p-adic integer is open. But i don't know how to prove that any finite index subgroup of multiplicative group of p-adic field is open using the above fact. Could someone give an idea for this?

Answer 1

A subgroup of index $n=p^r m$ contains $(1+p^2 O_K)^n$.

As $p\nmid m$, $(1+p^2 O_K)^m=1+p^2 O_K$.

Then you need to show that $$(1+p^2 O_K)^{p^r} = 1+p^{2+r} O_K$$ which is open.