Gauss Sums of the Cubic Character over $GF(2^m)$: an elementary derivation

TL;DR

This paper presents an elementary derivation of the Gauss sum for a cubic character over finite fields of characteristic two, providing explicit values without relying on the Davenport-Hasse theorem.

Contribution

It offers a novel elementary method to compute Gauss sums of cubic characters over GF(2^m), avoiding complex theorems.

Findings

Gauss sum is -1 when s is odd

Gauss sum is -(-2)^{s/2} when s is even

Elementary derivation simplifies understanding of Gauss sums

Abstract

An elementary approach is shown which derives the value of the Gauss sum of a cubic character over a finite field $\mathbb F_{2^s}$ without using Davenport-Hasse's theorem (namely, if $s$ is odd the Gauss sum is -1, and if $s$ is even its value is $-(-2)^{s/2}$).