Complete solution to cyclotomy of order 2l2 with prime l

• Published in: The Ramanujan journal Vol. 53; no. 3; pp. 529 - 550
• Main Authors: Ahmed, Md Helal, Tanti, Jagmohan, Hoque, Azizul
• Format: Journal Article
• Language: English
• Published: New York Springer US 2020; Springer Nature B.V
• Subjects: Combinatorics; Field Theory and Polynomials; Fourier Analysis; Functions of a Complex Variable; Mathematics; Mathematics and Statistics; Number Theory; 11T24; Cyclotomic numbers; Dickson–Hurwitz sums; 11T22; Jacobi sums; Cyclotomic fields
• ISSN: 1382-4090; 1572-9303
• DOI: 10.1007/s11139-019-00182-9

Let l ,  p be odd primes, q = p r , r ∈ Z + , q ≡ 1 ( mod 2 l 2 ) and F q a field with q elements. 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 but still effective formulae are yet to be known. In this paper we obtain an explicit expression for cyclotomic numbers of order 2 l 2 . The formula consists of the cyclotomic numbers of orders l , 2 l , l 2 and the coefficients of a special type Jacobi sum of order 2 l 2 . At the end, we illustrate the nature of two matrices corresponding to two types of cyclotomic numbers.