A tripling on the algebraic number field
type: Item
instance of: scholarly article
title: A tripling on the algebraic number field
zbMATH Open document ID: 0591.12013
author: Yasumasa Akagawa
published in: Osaka Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Let \(\ell\) be a prime number and let n be a power of \(\ell\). Let k be an algebraic number field of finite degree which contains a primitive n-th root of unity \(\zeta_ n\). In the present paper the author defines a tripling symbol \((x,y,z)_ n\in <\zeta_ n>\), which is defined on a subset of \(k^{\times}\times k^{\times}\times k^{\times}\) consisting of the elements called strictly orthogonal, and characterizes the norm group from the bicyclic extension \(k(^ n\sqrt{x},^ n\sqrt{y})\). In the case \(k={\mathbb{Q}}\) and \(n=2\), \textit{Y. Furuta} [Nagoya Math. J. 79, 79- 109 (1980; Zbl 0444.12001), cf. ibid. 93, 61-69 (1984; Zbl 0526.12009)] defined a similar symbol and studied it; though this symbol does not coincide with the above, these are essentially the same if these are defined at the same time. For each prime \({\mathfrak l}\) over (\(\ell)\) in k, fix a local Galois extension \(\Omega^{{\mathfrak l}}/k_{{\mathfrak l}}\) whose Galois group is a free pro-\(\ell\) group, where \(k_{{\mathfrak l}}\) is the completion of k by \({\mathfrak l}\). For \(x,y\in k^{\times}\), x and y are said to be orthogonal if its Hilbert symbol \((\frac{x,y}{{\mathfrak p}})_ n=1\) for every \({\mathfrak p}\). In the case \(\ell >2\), x and y are said to be strictly orthogonal if x and y are orthogonal and satisfy that \(x\in N_{k_{{\mathfrak l}}(^ n\sqrt{y})/k_{{\mathfrak l}}}(k_{{\mathfrak l}}(^ n\sqrt{y})^{\times}\cap (\) \(\Omega^{{\mathfrak l}^{\times}})^ n)\) and \(y\in N_{k_{{\mathfrak l}}(^ n\sqrt{x})/k_{{\mathfrak l}}}(k_{{\mathfrak l}}(^ n\sqrt{x})^{\times}\cap (\) \(\Omega^{{\mathfrak l}^{\times}})^ n)\). In the case \(\ell =2\), the notion of strictly orthogonal is defined similarly but needs some more conditions. Let x,y be \(\ell\)-independent and let x,y,z be pairwise strictly orthogonal. Let S be a set of primes of k containing the prime divisors of (\(\ell xyz)\infty\). Then there exists an element \(a\in k(^ n\sqrt{y})\) such that \(a^{l-\sigma}\equiv x\) mod …
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11S31
zbMATH DE Number: 3948419
zbMATH Keywords: tripling symbol
zbMATH Keywords: norm group
zbMATH Keywords: bicyclic extension
zbMATH Keywords: Galois group
zbMATH Keywords: free pro- \(\ell \) group
zbMATH Keywords: Hilbert symbol
zbMATH Keywords: strictly orthogonal
zbMATH Keywords: n-th power residue symbol
zbMATH Keywords: transgression relation
zbMATH Keywords: reciprocity law
zbMATH Keywords: norm theorem
zbMATH Keywords: class number
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3948419