International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 42818, 6 pages
doi:10.1155/IJMMS/2006/42818
Dickson curves
Department of Mathematics, The Pennsylvania State University, New Kensington Campus, New Kensington 15068, PA, USA
Received 6 March 2006; Accepted 26 March 2006
Copyright © 2006 Javier Gomez-Calderon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let kq
denote the finite field of order q
and odd
characteristic p. For a∈kq, let gd(x,a)
denote the
Dickson polynomial of degree d
defined by gd(x,a)=∑i=0[d/2]d/(d−i)(d−ii)(−a)ixd−2i. Let f(x)
denote a monic
polynomial with coefficients in kq. Assume that f2(x)−4
is not a perfect square and gcd(p,d)=1. Also assume that
f(x)
and g2(f(x),1)
are not of the form gd(h(x),c). In this note, we show that the polynomial gd(y,1)−f(x)∈kq[x,y]
is absolutely irreducible.