(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,  D_n(X, c)  and  E_n(X, c),  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(U, V) in Z[U, V] such that g(U+V, UV) = Uⁿ+Vⁿ,  and then D_n(X, c) = g(X, c).)

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  D_n(X, c)  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  F_p,  then the corresponding Brewer sum is the image in  F_p  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.

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