Explicit multiplicative relations between Gauss sums

Explicit Multiplicative Relations between Gauss Sums

H. Hasse conjectured that all multiplicative relations between Gauss sums essentially follow from the Davenport-Hasse product formula and the norm relation for Gauss sums. While this is known to be false, very few counterexamples, now known as sign ambiguities, have been given. Here, we provide an explicit product formula giving an infinite class of new sign ambiguities and resolve the ambiguou...

On Gauss-Jacobi sums

In this paper, we introduce a kind of character sum which simultaneously generalizes the classical Gauss and Jacobi sums, and show that this “Gauss-Jacobi sum” also specializes to the Kloosterman sum in a particular case. Using the connection to the Kloosterman sums, we obtain in some special cases the upper bound (the “Weil bound”) of the absolute values of the Gauss-Jacobi sums. We also discu...

Half Gauss Sums

Exponential Sums, Gauss Sums and Cyclic Codes

The dissertation consists of three articles in which the evaluation of certain exponential sums and Gauss sums and bounds for the absolute values of exponential sums are considered. The summary part of the thesis provides interpretations in terms of coding theory for the results obtained in the articles.

Multiplicative Order of Gauss Periods

We obtain a lower bound on the multiplicative order of Gauss periods which generate normal bases over finite fields. This bound improves the previous bound of J. von zur Gathen and I. E. Shparlinski.