There is a problem in Neukirch's Algebraic Number Theory,which is in Charpter $2$ section $5$ .
The problem is :
If $K$ is a $p-adic$ number field, then the groups $K^{*n}$, for $n$ belongs to $N$, form a basis of neighbourhoods of $1$ in $K^*$.
I think it is very clear that $K^{*n}$ is an open subgroup of $K^*$ for each integer $n$,but how to check they form a basis of neighbourhood of $1$ in $K^*$? This really puzzled me these days,so please offer me some help and show how to solve it,thank you very much!