How to find no. of polynomials over a finite field satisfying given condition. If i am given with a polynomial $$g(x) = \sum_{i=0}^4 a_ix^i$$ where $a_i \in \mathbb{F}_{2^k}$, finite field with $2^k$ elements. How to find out the no of such polynomials $g(x)$ such that either $g(w) = 0$ or $g(w^{-1}) = 0$, where $w$ is primitive $5^{th}$ root of unity. 

I believe the answer is: $1.$ Indeed, $q(x) = \sum_{i=0}^4 x^i$ is such a polynomial (for both $w$ and $1/w$), now let $g(x)$ be your polynomial, and consider $h(x)=g(x) - a_0 q(x)$ this will vanish at $w,$ but it is divisible by $x,$ and if $h(x)$ were not the zero polynomial, $w$ would satisfy an equation of degree less than four, which would make it not primitive.
