Abstract
Let l, p be odd primes, \(q=p^r,\) \(r\in \mathbf {Z}^{+},\) \(q\equiv 1\pmod {2l^2}\) and \(\mathbf {F}_q\) 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 \(2l^{2}.\) The formula consists of the cyclotomic numbers of orders \(\displaystyle {l, 2l, l^2}\) and the coefficients of a special type Jacobi sum of order \(2l^2.\) At the end, we illustrate the nature of two matrices corresponding to two types of cyclotomic numbers.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Acharya, V.V., Katre, S.A.: Cyclotomic numbers of orders \(2l, l\) an odd prime. Acta Arith. 69(1), 51–74 (1995)
Berndt, B.C., Evans, R.J., Williams, K.S.: Gauss and Jacobi Sums. Wiley, A Wiley-Interscience Publication, New York (1998)
Dickson, L.E.: Cyclotomy and trinomial congruences. Trans. Am. Soc. 37, 363–380 (1935)
Dickson, L.E.: Cyclotomy, higher congruences, and Waring’s problem. Am. J. Math. 57, 391–424 (1935)
Dickson, L.E.: Cyclotomy when \(e\) is composite. Trans. Am. Math. Soc. 38, 187–200 (1935)
Evans, R.J., Hill, J.R.: The cyclotomic numbers of order sixteen. Math. Comput. 33, 827–835 (1979)
Friesen, C., Muskat, J.B., Spearman, B.K., Williams, K.S.: Cyclotomy of order \(15\) over \(GF(p^{2})\), \(p\equiv 4, 11 ~(\text{ mod } \; 15)\). Int. J. Math. Math. Sci. 9, 665–704 (1986)
Katre, S.A., Rajwade, A.R.: Complete solution of the cyclotomic problem in \(\vec{F}^{*}_q\) for any prime modulus \({{l}}\), \(q=p^{\alpha }\), \(p\equiv 1 ~(\text{ mod } \; {{l}})\). Acta Arith. 45, 183–199 (1985)
Lehmer, E.: On the cyclotomic numbers of order sixteen. Can. J. Math. 6, 449–454 (1954)
Muskat, J.B.: The cyclotomic numbers of order fourteen. Acta Arith. 11, 263–279 (1966)
Muskat, J.B.: On Jacobi sums of certain composite orders. Trans. Am. Math. Soc. 134, 483–502 (1968)
Muskat, J.B., Whiteman, A.L.: The cyclotomic numbers of order twenty. Acta Arith. 17, 185–216 (1970)
Parnami, J.C., Agrawal, M.K., Rajwade, A.R.: Jacobi sums and cyclotomic numbers for a finite field. Acta Arith. 41, 1–13 (1982)
Shirolkar, D., Katre, S.A.: Jacobi sums and cyclotomic numbers of order \(l^{2}\). Acta Arith. 147, 33–49 (2011)
Storer, T.: On the unique determination of the cyclotomic numbers for Galois fields and Galois domains. J. Comb. Theory 2, 296–300 (1967)
Whiteman, A.L.: The cyclotomic numbers of order sixteen. Trans. Am. Math. Soc. 86, 401–413 (1957)
Whiteman, A.L.: The cyclotomic numbers of order twelve. Acta Arith. 6, 53–76 (1960)
Acknowledgements
The first and second authors thank Central University of Jharkhand, Ranchi, Jharkhand for providing necessary facilities to carry out this research.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The third author was supported by SERB-NPDF (PDF/2017/001958), Government of India.
Rights and permissions
About this article
Cite this article
Ahmed, M.H., Tanti, J. & Hoque, A. Complete solution to cyclotomy of order \(2l^{2}\) with prime l. Ramanujan J 53, 529–550 (2020). https://doi.org/10.1007/s11139-019-00182-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-019-00182-9