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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 $- {(- 2)}^{s / 2}$ ).

Incomplete character sums and a special class of permutations

S. D. Cohen, H. Niederreiter, I. E. Shparlinski, M. Zieve (2001)

Journal de théorie des nombres de Bordeaux

We present a method of bounding incomplete character sums for finite abelian groups with arguments produced by a first-order recursion. This method is particularly effective if the recursion involves a special type of permutation called an $ℛ$ -orthomorphism. Examples of $ℛ$ -orthomorphisms are given.

Jacobi sums and cyclotomic numbers of order l²

Devendra Shirolkar, S. A. Katre (2011)

Acta Arithmetica

Kloosterman sums and primitive elements in Galois fields

Stephen Cohen (2000)

Acta Arithmetica

Kloosterman sums as algebraic integers.

Benji Fisher (1995)

Mathematische Annalen

Linear congruences and a conjecture of Bibak

Chinnakonda Gnanamoorthy Karthick Babu, Ranjan Bera, Balasubramanian Sury (2024)

Czechoslovak Mathematical Journal

We address three questions posed by K. Bibak (2020), and generalize some results of K. Bibak, D. N. Lehmer and K. G. Ramanathan on solutions of linear congruences $\sum_{i = 1}^{k} a_{i} x_{i} \equiv b (mod n)$ . In particular, we obtain explicit expressions for the number of solutions, where $x_{i}$ ’s are squares modulo $n$ . In addition, we obtain expressions for the number of solutions with order restrictions $x_{1} \geq \dots \geq x_{k}$ or with strict order restrictions $x_{1} > \dots > x_{k}$ in some special cases. In these results, the expressions for the number of solutions involve Ramanujan...

On explicit formulas for the number of solutions to the equation ${(x_{1} + \dots + x_{n})}^{2} = a x_{1} \dots x_{n}$ in a finite field.

Baulina, Yu.N. (2003)

Sibirskij Matematicheskij Zhurnal

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Exponential sums on (Gm)n.

Fast computation of Gauss sums and resolution of the root of unity ambiguity

Further evaluations of Weil sums

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