Arithmetic of Finite Fields: First International Workshop, WAIFI 2007, Madrid, Spain, June 21-22, 2007. Proceedings

Explicit factorizations, into a product of irreducible polynomials, over Fq of thecyclotomic polynomials Q2n(x) are given in [4] when q ≡ 1 (mod 4). The caseq ≡ 3 (mod 4) is done in [5]. Here we give factorizations of Q2nr(x) where ris prime and q ≡ ±1 (mod r). In particular, this covers Q2n3(x) for all Fq ofcharacteristic not 2, 3. We apply this to get explicit factorizations of the firstand second kind Dickson polynomials of order 2n3 and 2n3 − 1 respectively.Explicit factorizations of certain Dickson polynomials have been used to computeBrewer sums [1]. But our basic motivation is curiosity, to see what factorsarise. Of interest then is how the generalized Dickson polynomials Dn(x, b) arisein the factors of the cyclotomic polynomials and how the Dickson polynomialsof the first kind appear in the factors of both kinds of Dickson polynomials.