On certain values of Kloosterman sums

Details On certain values of Kloosterman sums On certain values of Kloosterman sums

2009-03-25

arxiv.org/abs/0903.4419v3

Mathematics

Number Theory

4 pages; new proof which covers also the cases left open in ver.1; ver3: a typo corrected in Conj.2 and a small correction in

Scientific paper

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 in case $p>3$: if $a\ne0$ belongs to the proper subfield $\F_q$ of $\F_{q^n}$, then $K_{q^n}(a)\ne-1$. 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.

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