Cyoclotomic Numbers Of Order 2l^2 With Prime L

TL;DR

This paper derives explicit formulas for cyclotomic numbers of order 2l^2 using Jacobi sums, advancing the understanding of their structure and providing illustrative matrix representations.

Contribution

It provides a new explicit expression for cyclotomic numbers of order 2l^2 in terms of lower-order Jacobi sum coefficients.

Findings

Explicit formulas for cyclotomic numbers of order 2l^2

Representation of cyclotomic numbers via matrices

Connection between cyclotomic numbers and Jacobi sums

Abstract

The problem of determining cyclotomic numbers in terms of the solutions of certain Diophantine systems has been treated by many authors since the age of Gauss. In this paper we obtain an explicit expression for cyclotomic numbers of order 2l^2 in terms of the coefficients of the Jacobi sums of lower orders. At the end, we illustrate the nature of two matrices corresponding to two types of cyclotomic numbers.