On the number of solutions of some Kummer equations over finite fields
type: Item
instance of: scholarly article
title: On the number of solutions of some Kummer equations over finite fields
zbMATH Open document ID: 1372.11065
author: Florin Nicolae
published in: Finite Fields and their Applications
publication date: 2014-09-02T00:00:00Z
review text: Let \(k=\mathbb F_q\) be a finite field of characteristic \(p\) and order \(q\). Let \(\ell\neq p\) be another prime. The author computes the number \(N\) of solutions \((x,y)\) in \(k\) of the Kummer equation \[ y^{\ell}=x(x^{\ell^n}-1). \] For characters \(\varphi_1\), \(\varphi_2\) of \(k^\ast\), the Jacobi sum is: \[ \iota (\varphi_1, \varphi_2)=\-\sum_{c\in k} \varphi_1(c)\varphi_2(1-c). \] Let \(\zeta_m=\exp (2\pi i/m)\). The main result is: {\parindent=0.6cm\begin{itemize}\item[(a)] If \(q\not\equiv 1\pmod{\ell^{n+1}}\) then \(N=q\). \item[(b)] If \(q\equiv 1\pmod{\ell^{n+1}}\) then \(N=q-\mathrm{Tr}_{\mathbb Q(\zeta_{\ell^{n+1}})/\mathbb Q} (\eta_n)\), where \(\eta_n=\iota (\psi^{\ell^n},\psi)\) for a character \(\psi\) of \(k^*\) of order \(\ell^{n+1}\). \end{itemize}} (The paper gives the hypothesis of (a) as \(q\not\equiv 1\pmod{\ell}\) but that is a typographical error. The statement given here is what is proven.) Now \(|\eta_n|=\sqrt{q}\) and a further result determines \(\eta_n\) via its prime ideal factorization and a congruence in the ring \(\mathbb Z[\zeta_{\ell^{n+1}}]\).
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 11T24
zbMATH DE Number: 6338844
zbMATH Keywords: finite field
zbMATH Keywords: Kummer equation
zbMATH Keywords: Jacobi sum
MaRDI profile type: Publication
full work available at URL: j.ffa.2014.04.005
OpenAlex ID: W2032069344
cites work: Q3229467
cites work: Q4792305
cites work: Quotients of Fermat curves and a Hecke character
cites work: Q5690468
DOI: 10.1016/J.FFA.2014.04.005
description: scientific article; zbMATH DE number 6338844