On cyclotomic units connected with \(p\)-adic characters
type: Item
instance of: scholarly article
title: On cyclotomic units connected with \(p\)-adic characters
zbMATH Open document ID: 0547.12002
author: Tsuyoshi Uehara
published in: Journal of the Mathematical Society of Japan
publication date: 1985-01-01T00:00:00Z
review text: Let \(p\) be an odd prime and \(K\) an abelian number field containing any primitive \(p\)-th root \(\zeta\) of unity, of degree prime to \(p\). For an even character \(\chi\neq 1\) of the Galois group \(G\) of \(K\) over the rationals, we denote by \(\phi\),\({\bar\phi }\) the \(p\)-adic characters over \(\chi,\chi^{- 1}\omega\) respectively, where \(\omega\) is the character of \(G\) satisfying \(\zeta^{\omega (\sigma)}=\zeta^{\sigma}\) \((\sigma\in G)\). Here the values of any character are contained in an algebraic closure \(\Omega_p\) of the \(p\)-adic number field. For any abelian \(p\)-group \(S\) on which \(G\) acts we consider the \(\phi\)-part \(S_{\phi}\) and the \({\bar\phi}\)-part \(S_{{\bar\phi}}\) of \(S\). Let \(K_{\chi}\) be the fixed field of \(\mathrm{Ker }\chi\) and \(E_{\chi},C_{\chi}\) the unit group, the cyclotomic unit group of \(K_{\chi}\), respectively. A unit \(\eta_{\phi}\) in \(C_{\chi}\) is called a \(\phi\)-relative cyclotomic unit if \(\eta_{\phi} E^p_{\chi}\) generates \((C_{\chi} E^p_{\chi}/E^p_{\chi})_{\phi}\). Let \(m({\bar\phi}\)) be the exponent of the exact power of \(p\) dividing the Bernoulli number \(B_1(\chi\omega^{-1})\) in \(\Omega_p\). Further by \(A\) we mean the \(p\)-Sylow subgroup of the ideal class group of \(K\). One knows that the \(p^{m({\bar\phi})}\)-th power of any ideal representing a class in \(A_{{\bar\phi }}\) is principal. In the present paper it is proved that a \(\phi\)-relative cyclotomic mit \(\eta_{\phi}\) is a \(p\)-th power in \(K\) if and only if \(m({\bar\phi})>0\) and the \(p^{m({\bar\phi})}\)-th power of any ideal \({\mathfrak a}\), (\({\mathfrak a},p)=1\), representing a class in \(A_{{\bar\phi}}\), is generated by a \(p\)-primary integer in \(K\). Here an integer \(\alpha\) in \(K\) is said to be \(p\)-primary if \(K(\alpha^{1/p})/K\) is an unramified extension.
Mathematics Subject Classification ID: 11R20
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11S15
Mathematics Subject Classification ID: 11R21
zbMATH DE Number: 3871494
zbMATH Keywords: abelian number field
zbMATH Keywords: p-adic characters
zbMATH Keywords: cyclotomic unit
zbMATH Keywords: Bernoulli number
zbMATH Keywords: ideal class group
zbMATH Keywords: p-primary integer
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03710065
OpenAlex ID: W1992298834
DOI: 10.2969/JMSJ/03710065
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description: scientific article; zbMATH DE number 3871494