Multiplicative representation of algebraic numbers for the range of an arbitrary prime divisor
type: Item
instance of: scholarly article
title: Multiplicative representation of algebraic numbers for the range of an arbitrary prime divisor
zbMATH Open document ID: 46.0251.01
author: K. Hensel
published in: Journal für die Reine und Angewandte Mathematik
publication date: 1916-01-01T00:00:00Z
full work available at URL: https://www.digizeitschriften.de/dms/resolveppn/
full work available at URL: https://eudml.org/doc/149451
review text: Hensel begins with studying the unit group of the field of \(p\)-adic numbers \(\mathbb Q_p\). Every \(p\)-adic number is a power of \(p\) times a unit, and the unit group is isomorphic to \(\mathbb Z\) times a finite group of roots of unity. The residue classes modulo \(p^s\) of the units form a finite group of order \(\phi(p^s)\) whose structure is determined. In the second section, Hensel treats the corresponding problem for \(p\)-adic number fields, i.e., finite extensions \(K_\pi\) of \(\mathbb Q_p\). Hensel gives a basis for \(1\)-units (these are units \(\equiv 1 \bmod \pi\), where \(\pi\) generates the prime ideal of \(K_\pi\)) and he determines the roots of unity in \(K_\pi\). Editorial comment: Review added in 2016.
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11S15
Mathematics Subject Classification ID: 11S99
zbMATH DE Number: 2610417
zbMATH Keywords: \(p\)-adic numbers
zbMATH Keywords: unit groups
zbMATH Keywords: roots of unity
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2610417