Cyclotomic units over finite fields
type: Item
instance of: scholarly article
title: Cyclotomic units over finite fields
zbMATH Open document ID: 0872.11053
DOI: 10.1007/BF02849803
publication date: 1997-10-15T00:00:00Z
review text: The paper focusses on determining the size \(S\) of the image of the group of all cyclotomic units in \(\mathbb{Q}(\zeta_p)^+\) after reducing \(\mathbb{Z}[\zeta_p]^+\) modulo a prime \(q\neq p\). Knowing \(S\), some indices which naturally come up when studying units in integral group rings \(\mathbb{Z}[C]\) of cyclic groups \(C\) of order \(pq\) (see e.g. \textit{K. Hoechsmann} [Manuscr. Math. 75, 5--23 (1992; Zbl 0773.16016); Can. J. Math. 47, 113--131 (1995; Zbl 0827.16022)]) become accessible for computation. Having this application in mind, the paper starts out from a cyclic Galois algebra \(E\) over \(\mathbb{F}_q\) with group \(\langle\sigma\rangle\). Let \(\sigma^g\) act on \(E\) as the Frobenius automorphism \(e\mapsto e^q\) and have order \(f\). Then the group \(U\) of units in \(E\) is a free rank-one module for \(R=\mathbb{Z}[x]/\langle q^f-1,x^g-q\rangle\). Each \(v\in U\) gives rise to a defect \(d_q(v)= [U^{(j)}: Rv]\) with \(j=Nv\) and \(U^{(j)}= \{u\in U: Nu^j=1\}\), \(N=\sigma\)-norm. The main result gives the prime divisors of \(d_q(v)\). Of particular interest is a \(v\) such that \(Rv\) is the reduction modulo \(q\) of the \(p\)-th cyclotomic units.
Mathematics Subject Classification ID: 11T22
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 792845
zbMATH Keywords: finite fields
zbMATH Keywords: cyclotomic units
zbMATH Keywords: defect
zbMATH Keywords: prime divisors
author: Klaus Hoechsmann
reviewed by: Jürgen Ritter
MaRDI profile type: Publication
cites work: Q5522902
cites work: Constructing units in commutative group rings
cites work: Units in Integral Group Rings for Order pq
cites work: Q3943047
full work available at URL: https://doi.org/10.1007/bf02849803
OpenAlex ID: W2089544263
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 792845