Finite Fields Primitive and Irreducible Polynomials
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Serdica Journal of Computing, Vol. 2, No 3, (2008), 239p-248p
In this paper, we study the ratio θ(n) = λ2 (n) / ψ2 (n), where λ2 (n) is
the number of primitive polynomials and ψ2 (n) is the number of irreducible
polynomials in GF (2)[x] of degree n. Let n = ∏ pi^ri, i=1,..l
be the prime factorization of n. We show that, for fixed l and ri , θ(n) is close to 1 and θ(2n) is
not less than 2/3 for sufficiently large primes pi . We also describe an infinite
series of values ns such that θ(ns ) is strictly less than 1/2.
This work was presented in part at the 8th International Conference on Finite Fields and
Applications Fq^8 , Melbourne, Australia, 9-13 July, 2007.