Construction of a certain circular unit and its applications
type: Item
instance of: scholarly article
title: Construction of a certain circular unit and its applications
zbMATH Open document ID: 1229.11135
author: Jae Moon Kim
author: Jado Ryu
published in: Journal of Number Theory
publication date: 2011-03-03T00:00:00Z
review text: If \(K\) is number field and \(n\) a natural number, we denote by \(U(K)\) the unit group of the ring of integers of \(K\) and \(\zeta_n\) a primitive \(n\)-th root of unity. Let \(V_n\) be the subgroup of the multiplicative group \(\mathbb{Q}(\zeta_n)\setminus \{0\}\) generated by \(\pm \zeta_n\) and \(1-\zeta_n^{a}\), \(1 \leq a \leq n-1\). The group of cyclotomic units \(C(\mathbb{Q}(\zeta_n))\) of \(\mathbb{Q}(\zeta_n)\) is defined by: \(C(\mathbb{Q}(\zeta_n))=U(\mathbb{Q}(\zeta_n))\cap V_n\). Let \(K/\mathbb{Q}\) be a finite abelian extension having conductor \(n\) (by Kronecker-Weber theorem, the smallest natural number such that \(k\subset\mathbb{Q}(\zeta_n)\)). Let \(W_n\) be the subgroup of the multiplicative group \(\mathbb{Q}(\zeta_n)\setminus \{0\}\) generated by \(-1\) and the elements \(N_{\mathbb{Q}(\zeta_m)/K\cap \mathbb{Q}(\zeta_m)}(1-\zeta_m^{a})\), where \(m\not=1\) is a divisor of \(n\) relatively prime to \(a\). In [Invent. Math. 62, 181--234 (1980; Zbl 0465.12001)], \textit{W. Sinnott} defined the group \(C_S(K)\) of circular units of \(K\) by: \(C_S(K)=U(K)\cap W_n\). In [Introduction to cyclotomic fields. 2nd ed. Graduate Texts in Mathematics. 83. New York, NY: Springer (1997; Zbl 0966.11047)], \textit{L. Washington} defined \(C_W(K)=U(K)\cap C(\mathbb{Q}(\zeta_n))\). It is immediate that \(C_S(K)\) is a subgroup of \(C_W(K)\). Let \(n=p_1^{e_1}p_2^{e_2}\), where the \(p_i\), \(1 \leq i \leq 2\), are distinct prime odd numbers, and the \(e_i\), \(1 \leq i \leq 2\), are natural numbers. Let \(k/\mathbb{Q}\) be a finite real abelian extension with conductor \(n\). Assume that \(k\) is contained in the compositum of the maximal real subfields of \(\mathbb{Q}(\zeta_{p_1^{e_1} })\) and \(\mathbb{Q}(\zeta_{p_2^{e_2} })\) (\(\mathbb{Q}(\zeta_{p_1^{e_1}} +\zeta_{p_1^{e_1}}^{-1})\) and \(\mathbb{Q}(\zeta_{p_2^{e_2}} +\zeta_{p_2^{e_2}}^{-1})\) ). The main result of the paper is the following. The authors construct an element \(\theta\) in \(C_W(k)\) wi…
reviewed by: Bouchaïb Sodaïgui
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 5859977
zbMATH Keywords: circular units
zbMATH Keywords: index formula
zbMATH Keywords: class number
MaRDI profile type: Publication
full work available at URL: j.jnt.2010.11.002
OpenAlex ID: W2027042125
cites work: On bases of groups of circular units of some imaginary abelian number fields
cites work: A note on Sinnott's index formula
cites work: Central Extensions, Galois Groups, and Ideal Class Groups of Number Fields
cites work: On relatively invariant circular units and Stickelberger elements
cites work: On the parity of the class number of a biquadratic field
cites work: On the Stickelberger ideal and the circular units of an abelian field
cites work: Q3874309
DOI: 10.1016/J.JNT.2010.11.002
description: scientific article; zbMATH DE number 5859977