Gauss Sums of Cubic Characters over $\mathbb F_{p^r}$, $p$ Odd
Volume 60 / 2012
Bulletin Polish Acad. Sci. Math. 60 (2012), 1-19
MSC: Primary 11T24; Secondary 11T06.
DOI: 10.4064/ba60-1-1
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. 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 $p$ 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.
