cp's OEIS Frontend

Comment (entirely taken from Cugiani's text - see References) from Vincenco Librandi, Aug 23 2011: (Start)

This deals with primitive polynomials in GF_k(p). There are p^k monic k-th order polynomials J(p) = x^k + a(k-1)*x^(k-1) + ... + a(0), because there are k independent coefficients a(.), each restricted modulo the prime p. phi(p^k-1)/k of these polynomials are primitive, where phi=A000010. [Example for p=7 and k=2: phi(7^2-1)/2 = phi(48)/2 = 16/2=8. See A011260 for p=2, A027385 for p=3, A027741 for p=5 etc.] Of these sets of primitive polynomials we select with p=1031 the polynomial x^3+73*x^2+x+67 for k=3 in A163303 and x^4+984*x^3+90*x^2+394-x+858 for k=4 in A163304 by the following criteria (This could be extended to k=5, 6,...): Let r = (p^k -1)/(p-1). We demand (see Theorem 1 in Hansen-Mullen)

i) (-1)^k a(0) is a primitive element of J(p).

ii) The remainder of the division of x^r through the polynomial equals (-1)^k a(0).

iii) The remainder of the division of x^(r/q) through the polynomial must have positive degree for each prime divisor q|r.

(End)