Gauss sums of cubic characters over $GF(p^r)$, $p$ odd

TL;DR

This paper presents an elementary method to compute Gauss sums of cubic characters over finite fields of odd characteristic, revealing new relationships between sums over different extensions and connections to prime representations.

Contribution

It introduces a novel elementary approach to evaluate Gauss sums without Davenport-Hasse's theorem and uncovers new links between sums over various field extensions.

Findings

Derived explicit values of Gauss sums for cubic characters over GF(p^r)

Established new relations between Gauss sums across field extensions

Connected prime representations to quadratic forms in cyclotomic fields

Abstract

An elementary approach is shown which derives the values of the Gauss sums over $\mathbb F_{p^r}$, $p$ odd, of a cubic character without using Davenport-Hasse's theorem. New links between Gauss sums over different field extensions are shown in terms of factorizations of the Gauss sums themselves, which are then rivisited in terms of prime ideal decompositions. Interestingly, one of these results gives a representation of primes $p$ of the form $6k+1$ by a binary quadratic form in integers of a subfield of the cyclotomic field of the $p$-th roots of unity.