  Let GF(q), q=p^r, be a finite field with a primitive element g. In this paper
we use exponential sums and Jacobi sums to compute the number of the
irreducible polynomials of degree m over GF(q) with trace fixed and norm
restricted to a coset of a subgroup <g^s>, s|(q-1). We give the number
explicitly for s=2, 3, 4 when q=p, and for s|(p^e+1) when r=2en. Finally, we
give explicit formulae for the number when both trace and norm are fixed, p=2
and m =< 30.
