On the number of primitive polynomials over finite fields

Let GF(q) denote the finite field of order q, a power of a prime p, and m a positive integer. Let Pqα(m) denote the number of primitive polynomials of degree m over GF(q) with trace α, where α GF(q). Cohen (Discrete Math. 83 (1990) 1; Lecture Notes Pure Appl. Math. 141 (1993)) proved that Pqα(m) is positive except for the cases P40(3)=Pq0(2)=0. In this paper, we provide several results on the enumeration problem of Pqα(m). Especially, we give several sufficient conditions for which Pqα(m) is constant for any nonzero trace α GF(q).