Let $p$ be an odd prime number.
As a corollary of Hensel's lemma, we know that for $m\in \mathbb{N}$ there is a primitive $m$-th root of unity iff $m|(p-1)$.
Hence, if $p\equiv 1$ mod 4, we have a $2^n$-th root of unity in $\mathbb{Q}_p$, for some $n\geq 2$.
My question is the following: once we have established the existence of such roots, how can we calculate the exact number of 2-power roots of unity in the $p$-adic field?
For $p$ an odd prime, the roots of unity in $\Bbb Q_p$ correspond to the nonzero elements of $\Bbb F_p$ and form a cyclic group of order $p-1$. So the number of $2$-power roots of unit in $\Bbb Q_p$ is the largest power of $2$ dividing $p-1$.