Complete solution to cyclotomy of order \(2l^2\) with prime \(l\)
The study of cyclotomic numbers and Jacobi sums have an important and a special place in number theory. In this article, the authors have established a complete explicit expression of cyclotomic numbers for order \(2l^{2}\) where \(l\) is an odd prime, in terms of, the lower orders and Jacobi sums. Let \(e \ge 2\) be an integer, \(p\) a rational prime, \(q = p^{r}\),\(r \in Z^{+}\) and \(q \equiv 1 \pmod e\). Let \(F_{q}\) be a finite field of \(q\) elements. We can write \(q = p^{r} = ek + 1\) for some \(k \in Z^{+}\). Let \(\gamma\) be a generator of the cyclic group \(F^{\ast}_{q}\) and \(\zeta _{e} = exp(2\pi i/e)\). Also for \(a \in F^{\ast}_{q}\) , ind\(_{\gamma} (a)\) is defined to be a positive integer \(m \le q - 1\) such that \(a = \gamma ^{m}\). Define a multiplicative character \(\chi_{e} : F^{\ast}_{q} \longrightarrow Q(\zeta _{e})\) by \(\chi_{e}(\gamma ) = \zeta _{e}\) and extend it on \(F_{q}\) by putting \(\chi _{e}(0) = 0\). For integers \(0 \le i, j \le e-1\), the Jacobi sum \(J_{e}(i, j)\) is defined by \[J_{e}(i, j) = \Sigma _{v\in F_{q}}\chi _{e}^{i}(v)\chi ^{j}_{e} (v + 1)\]. For \(0 \le i, j \le e - 1\), the cyclotomic numbers \((i, j)_{e}\) of order \(e\) are defined as follows: \((i, j)e:= \# \{ v \in F_{q} |\chi _{e}(v) = \zeta ^{i}_{e} , \chi_{e}(v + 1)= \zeta ^{j}_{e} \} \\ = \# \{ v \in F_{q} \setminus \{0, -1\} \ | \ ind_{\gamma} v \equiv i \pmod e, ind_{\gamma} (v +1) \equiv j \pmod e\}.\) The cyclotomic numbers \((i, j)_{e}\) and the Jacobi sums \(J_{e}(i, j)\) are well connected by the following relations \[ \Sigma_{i} \Sigma_{j} (i, j)_{e}\zeta ^{ai+bj }_{e} = J_{e}(a, b),\] and \[ \Sigma_{i} \Sigma_{j}\zeta ^{-(ai+bj)} _{e} J_{e}(i, j) = e^{2}(a, b)_{e}. \] The main theorem gives a very lengthy equation for the cyclotomic number \(4 l^{2}(a,b)_{2l^{2}}\) in terms of cyclotomic numbers of orders \(l^{2}, \ 2l\) and \(l\). The proof of the theorem uses the second equation connecting cyclotomic numbers and Jacobi sums repeatedly and some calculations on Jacobi sums of order \(2l^{}\). Also, there is a discussion on the relations between Jacobi sums and Dickson-Hurwitz sums. At the end of the article, the authors discuss the matrix associated with cyclotomic numbers for two particular values of \(p\). The article has only a moderate number of references so that those who read this article can have a better understanding of the results cited there. This article would be an interesting article for number theorists.
scientific article; zbMATH DE number 7345285
scientific article