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## Gauss sums of cubic character over {\$\Bbb F_{p^r},\ p\$}} odd

 Type of publication Peer-reviewed Original article (peer-reviewed) Schipani D., Elia M., New Cryprosystems based on Algebra
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### Original article (peer-reviewed)

 Journal Bull. Pol. Acad. Sci. Math. 60(1) 1 - 19 Bull. Pol. Acad. Sci. Math. 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 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.
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