Manjul Bhargava and Michael E. Zieve:
Factoring Dickson polynomials over finite fields,
Finite Fields Appl. 5 (1999), 103–111. MR 2000d:11151

(The published version is available online.)

In a recent paper, Chou described the irreducible factors of any Dickson polynomial over any finite field. We give a simpler statement and proof of these results. Here, for any element  c  of a field  K,  and any positive integer  n,  the Dickson polynomials of the first and second kind,  Dn(Xc)  and  En(Xc),  are the unique polynomials in  K[X]  which map  Y+(c/Y)  to  Yn+(c/Y)n  and  (Yn+1-(c/Y)n+1) / (Y-(c/Y)),  respectively. (The existence of these polynomials follows from the symmetric function theorem: for instance, this theorem implies there is a polynomial g(UV) in Z[UV] such that g(U+VUV) = Un+Vn,  and then Dn(Xc) = g(Xc).)

We also describe the irreducible factors of  g(X)-g(Y)  over a finite field, where  g(X)  is a Dickson polynomial of the first kind.

Additional comment added May 2006:  the factorizations of  Dn(Xc)  over a finite field were used by Alaca to prove that certain Brewer sums do not vanish; here, if  g(X)  is a fixed Dickson polynomial of the first kind over the prime field  Fp,  then the corresponding Brewer sum is the image in  Fp  of the sum, over all integers  i = 0, 1, ..., p-1,  of the Legendre symbols of  g(i)  modulo  p.

Additional comment added November 2007:  alternate descriptions of the irreducible factors of the Dickson polynomials are given by Fitzgerald and Yucas in papers from 2005, 2007, and 2007.


Michael Zievehome page   publication list