Trace formulae for irreducible polynomials over \(\mathbb F_P\) with minimal order roots in \(\mathbb F_{P^q}\) (Q958611)

Let \(P\) be a prime of the form \(P = q^ns+1\) for a prime \(q\). Then \(P^q = q^{n+1}sK+1\), where \(\gcd(K,P-1) = 1\). The author gives formulas involving values of the trace function \(\text{Tr}: \mathbb F_{P^q}\to\mathbb F_P\) of elements \(\alpha \in \mathbb F_{P^q}\) of order \(R\) for a prime divisor \(R\) of \(K\). For instance \(\text{Tr}(\alpha)+\text{Tr}(\alpha^{-1}) = -1\), \(\text{Tr}(\alpha)\text{Tr}(\alpha^{-1}) = (q+1)/2\) if \(q > 2, R = 2q+1\) (take e.g. \(P=401\), \(q=5\), \(R=11\)).