On the index of the group generated by relative units
type: Item
instance of: scholarly article
title: On the index of the group generated by relative units
zbMATH Open document ID: 0802.11043
author: Yoshitaka Odai
published in: Journal of Number Theory
publication date: 1994-12-15T00:00:00Z
review text: For an algebraic number field \(K\), denote by \(E_ K\) and \(W_ K\) the group of units and the group of roots of unity in \(K\), respectively. Consider a fixed abelian extension \(L\) of degree \(n\) over \(K\), with \(G= \text{Gal} (L/K)\). Let \(M\) be an intermediate field which is cyclic over \(K\). This article deals with the relative units of \(M/K\), i.e., the elements \(\varepsilon\in E_ M\) for which the norm \(N_{M/F} \varepsilon\) belongs to \(W_ F\) whenever \(F\) is an extension of \(K\) properly contained in \(M\). Let \(E_{M/K}\) denote the group of these relative units and write \({\mathcal E}= \prod_ M E_{M/K} W_ L/W_ L\), the product being over all possible \(M\). The author finds upper bounds for the index \([E:{\mathcal E}]\), where \(E= E_ L/ W_ L\). His previous results [Proc. Jap. Acad., Ser. A 64, 304-306 (1988; Zbl 0672.12002)] showed that this index divides \(n^ r\), where \(r\) is the \(\mathbb{Z}\)-rank of \(E\). Now he introduces a group \({\mathbf E}\) with \({\mathcal E}\leq {\mathbf E} \leq E\) and majorizes the indices \([{\mathbf E}:{\mathcal E}]\) and \([E:{\mathbf E}]\) separately. The group \({\mathbf E}\) depends on the decomposition of the \(\mathbb{Q} [G]\)-module \(E\otimes_ \mathbb{Z} \mathbb{Q}\) through the \(\mathbb{Q}\)- irreducible characters of \(G\); this allows one to reduce the study to the individual components thus obtained.
Mathematics Subject Classification ID: 11R20
Mathematics Subject Classification ID: 11R27
zbMATH DE Number: 556372
zbMATH Keywords: abelian extension
zbMATH Keywords: relative units
zbMATH Keywords: upper bounds
zbMATH Keywords: index
MaRDI profile type: Publication
full work available at URL: jnth.1994.1004
OpenAlex ID: W2031808997
DOI: 10.1006/JNTH.1994.1004
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description: scientific article; zbMATH DE number 556372