Thanks to some assistance below, I can now
show that if $g(X) \in \Bbb F_2[X]$ then $g(X)^2 = g(X^2)$.
Is there some more direct way to prove this special case (not that the original proof is long):
Show that if $\beta$ is a zero of $P(X) \in \Bbb F_2[X]$ then so it $\beta^2$.
Further, suppose that $\beta$ is a $N$th root of unity. If $N$ is not of the form $N = 2^d - 1$, then surely we can generate all powers $\beta^j$ with some power $n$, ie $\beta^n = \beta^{j + kN}$ for some integer $k$. (This is not the case for $N = 2^d - 1$ since we repeat: $\beta^{2^d} = \beta^{N+1} = \beta$.) Is this correct?
As always, any advice, muchos appreciato!
HINT: $$(a_1 + \dots + a_n)^2 = a_1^2 + \dots + a_n^2 + 2 (a_1a_2 + \dots + a_{n-1}a_n)$$ What happens if $2=0$?