For a local field $K$, I know that $c:=(K^* : (K^*)^m) = mq^{v(m)}\mid\mu_m(K)\mid$,
where $q$ is the number of residue field of $K$.
I want to know the quantity $c$ when $m=2$.
Using the above formula, if $v \nmid 2$, then $c=4$.
But how to compute for $v\mid2$?
