On unramified Galois extensions constructed using Galois representations
type: Item
instance of: scholarly article
title: On unramified Galois extensions constructed using Galois representations
zbMATH Open document ID: 1010.11064
author: Hiroyuki Hasebe
published in: Manuscripta Mathematica
publication date: 2002-12-01T00:00:00Z
review text: Let \(k\) be an algebraic number field and \(k_{\text{ab}^r}\) the maximal abelian extension of \(k_{\text{ab}^{(r-1)}}\), \(r \in\mathbb{N}\). Suppose that we have an \(n\)-dimensional Galois representation \(\rho\) of \(k\) such that the restriction of \(\rho\) to each decomposition group over a finite set of primes of \(k\) (containing \(\infty\) and the primes at which \(\rho\) ramifies) is triangulable. Let the field \(K\) correspond to the kernel of \(\rho\) and \(2^{r-2} < n \leq 2^{r-1}\). Then it is proved that \(K k_{\text{ab}^r}\) is unramified over \(k_{\text{ab}^r}\). This result is applied to ordinary Galois representations of \(\mathbb{Q}\) to obtain unramified extensions \(K \mathbb{Q}_{\text{ab}^2}\) over \(\mathbb{Q}_{\text{ab}^2}\).
reviewed by: Günter Lettl
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 11R21
zbMATH DE Number: 1837469
zbMATH Keywords: maximal abelian extension
zbMATH Keywords: \(n\)-dimensional Galois representation
zbMATH Keywords: ordinary Galois representations
zbMATH Keywords: unramified extensions
MaRDI profile type: Publication
full work available at URL: s002290200282
OpenAlex ID: W1983000632
DOI: 10.1007/S002290200282
description: scientific article; zbMATH DE number 1837469