I just got done with the proof of the following theorem:
Let $(K, |\cdot|)$ a locally compact nonarchimedean field with normalized discrete valuation $v$ and residue field $\mathbb{F}_q$. Let $m$ a positive integer not divisible by the characteristic of $K$, and let $\mu_m(K)$ the $m$-th roots of unity in $K$. Let $U$ be the unit group of $K$ (which is $v^{-1}(0)$), and let $U^n$ be the $n$-th higher unit group $1 + \mathfrak m^n$, where $\mathfrak m$ is the maximal ideal of the valuation ring $R$. Then
$$[U^* : U^{*m}] = q^{v(m)}\cdot \# \mu_m(K)$$
The following result is supposed to be a corollary of the above theorem, but I do not see why that is the case:
Let $(K, v, \pi, k)$ a locally compact field of characteristic $\neq 2$. Here $v$ is a valuation, $\pi$ is a uniformizer, and $k$ is the residue field. If the characteristic of $k > 2$, there are exactly $3$ quadratic extensions of $K$. If the characteristic of $k = 2$, there are exactly $2^{[K : \mathbb{Q}_p] + 2}-1$ quadratic extensions of $K$.
How can I see this?