Jacobi Sums and Cyclotomic Numbers of Order 2l2

Abstract

The dominant theme of this thesis is the determination of Jacobi sums and cyclotomic numbers of order 212 with l gt 3 a prime. The Jacobi sums and cyclotomic numbers have incredible applications in various fields, such as coding theory, cryptosystems, primality testing, difference sets and so forth. newlineLet p be a prime number and q = pr, r E Z. Let Fq be a finite field of q elements and 7 a generator of the cyclic group F. Determination of Jacobi sums and cyclotomic numbers of a particular order in a finite field Fq in terms of a solution of certain Diophantine system has been considered by many authors. But most of the solutions were having sign ambiguity. It was not clear that which solution corresponds to the generator -y. To resolve this ambiguity, many authors attempted the problem in terms of Jacobsthal sums, quadratic partitions, Dickson-Hurwitz sums and so forth. Later, the study of cyclotomic problems extended to prime orders, twice of prime or prime square orders etc. Along with this, many authors also investigated the relations among cyclotomic numbers (respectively Jacobi sums) with a view to reducing the complexity in the evaluation of all cyclotomic numbers (respectively Jacobi sums). Another problem that emerges in the literature is to evaluate Jacobi sums by utilizing their congruences, prime ideal decompositions and absolute values. Prime ideal decompositions and absolute values of Jacobi sums are already in the literature, thus the primary interest is to give the congruences for Jacobi sums. newline