If $k \in \mathbb{N}, n={2^k} $. Probe that $w$ is a primitive nth-root of unity $\iff$ $w$ is a root of $ P_k = x^{2^{k-1}} + 1 $

Question

Going to the right is understandable.

$w$ is a $2^k $-th primitive root of unity $\implies$ $w$ is a root of $P_k = x^{2^{k-1}} $

1. $ w \in G_{2^k} \implies w^{2^{k}} = 1$
2. But then $w$ is a primitive root, so $w^{2^{k-1}}$ $ \neq $ 1.
3. $(w^{2^{k-1}})^2 = w^{2^{k}} = 1$
4. $(w^{2^{k-1}}) =$ 1 or -1
5. Can't be 1 because of (2).
6. Then $w^{2^{k-1}} = -1 $

But I have trouble with going the other way and reversing the implication. Any help?

Answer 1

Proving that $w$ is a root of unity if $P_k=0$ is relatively easy: simply multiply both sides of $0=w^{2^{k-1}}+1$ by $w^{2^{k-1}}$. Proving that it is a primitive root of unity is a bit more difficult, but here is a hint: if $w^m=1$ for some $m<2^k$, then $m$ must be a power of 2.