The index of multiplicative group of local field

Question

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 $v\mid2$?