On Certain Values of Kloosterman Sums
Let K(q)n (a) be a Kloosterman sum over the finite field F(q)n of characteristic p. In this note so called subfield conjecture is proved: if a not equal 0 belongs to the proper subfield F(q) of F(q)n, then K(q)n (a) not equal -1. This completes recent works on the subfield conjecture by Shparlinski, and Moisio and Lisonek. The problem is motivated by some applications to bent functions. Moreover, in the course of the proof a large class of translates of Dickson polynomials are shown to be irreducible.
kloosterman sum dickson polynomial lucas number primitive divisor