On certain values of Kloosterman sums
Marko Moisio

TL;DR
This paper proves the subfield conjecture for Kloosterman sums over finite fields of characteristic greater than 3, showing that certain sums do not equal -1 when the element belongs to a proper subfield, with implications for bent functions.
Contribution
The paper proves the subfield conjecture for Kloosterman sums in characteristic p>3 and demonstrates the irreducibility of a large class of Dickson polynomial translates.
Findings
Confirmed the subfield conjecture for p>3
Established irreducibility of certain Dickson polynomial translates
Extended previous results by Shparlinski, Moisio, and Lisonek
Abstract
Let be a Kloosterman sum over the finite field of characteristic . In this note so called subfield conjecture is proved in case : if belongs to the proper subfield of , then . 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.
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Taxonomy
TopicsAnalytic Number Theory Research · Finite Group Theory Research · Advanced Algebra and Geometry
