I am going through a paper, and theres one point in the proof I've been having problems understanding.
We have $\alpha \in G$ with minimal polynomial $h$, where $G$ is the algebraic closure of a field $F$, a finite field of prime order $q$. Also take $[F(\alpha):F]=d$, and take $P_q(n)$ as the set of all monic polynomials over the field of order $q$ of order $n$. We have $1\leq k.
I'm having problems showing that: $[ \prod_{i=1}^n(x-x^{q^i})]^k \cdot \sum_{f\in P_q(n), h \nmid f}\frac{1}{f^k} \equiv 0 \pmod {(x-\alpha)^{k\lfloor \frac{n}{d} \rfloor}}$