Elliptic curves and explicit enumeration of irreducible polynomials with two coefficients prescribed

Let F-q be a finite field of characteristic p = 2, 3. We give the number of irreducible polynomials x(m) + a(m-1) x(m-1) + ... + a(0) F-q [x] with a(m-1) and a(m-3) prescribed for any given in if p = 2, and with a(m-1) and a(1) prescribed for m = 1,.... 10 if p = 2, 3. In the latter case an enumeration formula, applicable also if m > 10, is given, but it is explicit only up to the evaluation of certain Kronecker class numbers. (c) 2008 Elsevier Inc. All rights reserved.