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An elementary approach is shown which derives the values of the Gauss sums over $_{p^{r}}$ , 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 pth roots of unity.

Gauss Sums of the Cubic Character over $G F (2^{m})$ : an Elementary Derivation

Davide Schipani, Michele Elia (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

By an elementary approach, we derive the value of the Gauss sum of a cubic character over a finite field $_{2^{s}}$ without using Davenport-Hasse’s theorem (namely, if s is odd the Gauss sum is -1, and if s is even its value is $- {(- 2)}^{s / 2}$ ).

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Gauss sum for the adjoint representation of $G L_{n} (q)$ and $S L_{n} (q)$

Gauss sums for O⁺(2n,q)

Gauss sums for O⁻(2n,q)

Gauss sums for orthogonal groups over a finite field of characteristic two

Gauss sums for prime powers in p-adic fields

Gauss Sums for Symplectic Groups Over a Finite Field.

Gauss Sums of Cubic Characters over $_{p^{r}}$ , p Odd

Gauss Sums of the Cubic Character over $G F (2^{m})$ : an Elementary Derivation

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