(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(X, c) and En(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 Dn(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 Dn(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 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.
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