Irreducible Polynomials over GF(2) with Three Prescribed Coefficients

For an odd positive integer n, we determine formulas for the number of irreducible polynomials of degree n over GF (2) in which the coefficients of xn−1, xn−2 and xn−3 are specified in advance. Formulas for the number of elements in GF (2n) with the first three traces specified are also given. Let q be a prime power and let GF (q) be a finite field with q elements. A classical result (see [6, 3.25]) gives the number, Pq(n), of monic, irreducible polynomials of degree n over GF (q): Pq(n) = 1 n ∑ d|n μ(d)q, where μ is the Möbius function. This has been refined several times by counting the number Pq(n, 21, 22, . . . , 2k) of monic irreducible polynomials over GF (q) with the first k coefficients being the prescribed values 21, . . . , 2k. We are writing polynomials here as p(x) = x + a1xn−1 + a2xn−2 + · · ·+ an−1x + an. Carlitz [1] gave a formula for Pq(n, 21). Kuz’min [5] extended this to a formula for Pq(n, 21, 22). This was re-discovered, for the case q = 2, in [2] which also introduced the connection with higher traces. The same connection was used in [8] to get a formula for Pq(n, 21, 22, 23) when q = 2 and n is even. We complete this case, getting a formula for Pq(n, 21, 22, 23) when q = 2 and n is odd. The proof is quite different and depends on computations with quadratic forms. The higher traces are defined as follows. Let F be any field and let K/F be a separable extension of degree n. Let σ0, . . . , σn−1 be the monomorphisms from K into the algebraic closure of F . Then define for α ∈ K: tr1(α) = n−1 ∑ i=0 σi(α) tr2(α) = ∑ 0≤i<j≤n−1 σi(α)σj(α) tr3(α) = ∑ 0≤i<j<k≤n−1 σi(α)σj(α)σk(α)