$m$-th primitive root of unity over $\mathbb{Z}_{2^k}$ for some integer $k$

Question

Can we find the $m$-th primitive root of unity over $\mathbb{Z}_{2^k}$ when $m$ is also a power of $2$?

Answer 1

We have $\mathbb{Z}_{2^k}^\times \cong C_2 \times C_{2^{k-2}}$ and so $\mathbb{Z}_{2^k}^\times$ has a primitive $m$-th root of unit iff $m \mid 2^{k-2}$.

In this case, $5^{\frac{2^{k-2}}{m}}$ is a primitive $m$-th root of unit, since $5$ has order $2^{k-2}$ mod $2^k$.