Gauss sums of cubic characters over $\mathbb{F}_{p^{r}}$, $\mathit{p}$ odd

Abstract

An elementary approach is shown which derives the values of the Gauss sums over $\mathbb{F}_{p^{r}}$, $\mathit{p}$ odd, of a cubic character. New links between Gauss sums over different field extensions are shown in terms of factorizations of the Gauss sums themselves, which are then revisited in terms of prime ideal decompositions. Interestingly, one of these results gives a representation of primes $\mathit{p}$ of the form 6$\mathit{k}$+1 by a binary quadratic form in integers of a subfield of the cyclotomic field of the $\mathit{p}th roots of unity.