Complete solution to cyclotomy of order $$2l^{2}$$ 2 l 2 with prime l
- PDF / 339,746 Bytes
- 22 Pages / 439.37 x 666.142 pts Page_size
- 50 Downloads / 162 Views
Complete solution to cyclotomy of order 2l 2 with prime l Md Helal Ahmed1 · Jagmohan Tanti1
· Azizul Hoque2
Received: 10 December 2018 / Accepted: 7 June 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract Let l, p be odd primes, q = pr , r ∈ Z+ , q ≡ 1 (mod 2l 2 ) and Fq 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 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. Keywords Jacobi sums · Cyclotomic numbers · Dickson–Hurwitz sums · Cyclotomic fields Mathematics Subject Classification 11T22 · 11T24
1 Introduction Let e ≥ 2 be an integer, p a rational prime, q = pr , r ∈ Z+ and q ≡ 1 (mod e). Let Fq be a finite field of q elements. We can write q = pr = ek + 1 for some k ∈ Z+ . Let γ be a generator of the cyclic group Fq∗ and ζe = exp(2πi/e). Also for a ∈ Fq∗ , indγ (a) is defined to be a positive integer m ≤ q − 1 such that a = γ m . Define a
The third author was supported by SERB-NPDF (PDF/2017/001958), Government of India.
B
Jagmohan Tanti [email protected] Md Helal Ahmed [email protected] Azizul Hoque [email protected]
1
Department of Mathematics, Central University of Jharkhand, Ranchi 835 205, India
2
Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India
123
M. H. Ahmed et al.
multiplicative character χe : Fq∗ −→ Q(ζe ) by χe (γ ) = ζe and extend it on Fq by putting χe (0) = 0. For integers 0 ≤ i, j ≤ e −1, the Jacobi sum Je (i, j) is defined by Je (i, j) =
j
χ e i (v)χe (v + 1).
v∈Fq
However in the literature a variation of Jacobi sums is also considered and is defined by j
Je (χei , χe ) =
j
χei (v)χe (1 − v),
v∈Fq j
but are related by Je (i, j) = χei (−1)Je (χei , χe ). For 0 ≤ i, j ≤ e − 1, the cyclotomic numbers (i, j)e of order e are defined as j follows: (i, j)e :=#{v ∈ Fq |χe (v) = ζei , χe (v + 1) = ζe } = #{v ∈ Fq \ {0, −1} | indγ v ≡ i (mod e), indγ (v + 1) ≡ j (mod e)}. The cyclotomic numbers (i, j)e and the Jacobi sums Je (i, j) are well connected by the following relations [2,14]: i
ai+bj
(i, j)e ζe
= Je (a, b),
(1)
j
and i
−(ai+bj)
ζe
Je (i, j) = e2 (a, b)e .
(2)
j
(1) and (2) show that if we want to calculate all the cyclotomic numbers (i, j)e of order e, it is sufficient to calculate all the Jacobi sums Je (i, j) of the same order, and vice versa. Cyclotomic numbers are one of the most important objects in number theory and in other branches of mathematics. These numbers have been extensively used in coding theory, cryptography and in other branches of information theory. One of the central problems in the study of these num
Data Loading...
