Kloosterman sums, elliptic curves, and irreducible polynomials with prescribed trace and norm

TL;DR

This paper investigates the number of irreducible polynomials with specific trace and norm in finite fields, improves bounds on element counts, and explores properties of Kloosterman sums, providing new bounds and proofs.

Contribution

It offers new bounds for irreducible polynomials with prescribed coefficients and characterizes Kloosterman sums, extending previous results and simplifying existing proofs.

Findings

Improved bounds for the number of irreducible polynomials with prescribed trace and norm.

Sharp bounds for elements in finite fields with given trace and norm.

Characterization of Kloosterman sums divisible by three over ^r.

Abstract

Let $\F_q$ ($q=p^r$) be a finite field. In this paper the number of irreducible polynomials of degree $m$ in $\F_q[x]$ with prescribed trace and norm coefficients is calculated in certain special cases and a general bound for that number is obtained improving the bound by Wan if $m$ is small compared to $q$. As a corollary, sharp bounds are obtained for the number of elements in $\F_{q^3}$ with prescribed trace and norm over $\F_q$ improving the estimates by Katz in this special case. Moreover, a characterization of Kloosterman sums over $\F_{2^r}$ divisible by three is given generalizing the earlier result by Charpin, Helleseth, and Zinoviev obtained only in the case $r$ odd. Finally, a new simple proof for the value distribution of a Kloosterman sum over the field $\F_{3^r}$, first proved by Katz and Livne, is given.