Classification and prime decomposition of abelian \(q\)-fields
type: Item
instance of: scholarly article
title: Classification and prime decomposition of abelian \(q\)-fields
zbMATH Open document ID: 0888.11042
published in: Chinese Science Bulletin
publication date: 1998-06-07T00:00:00Z
review text: The author considers abelian \(q\)-fields, \(L\), which are defined to be Galois extensions of the rational number field with Galois group an abelian \(p\)-group. Such a field can be written as the compositum of subfields \(K_1,K_2,\dots, K_n\), where each \(K_i\) is a cyclic \(q\)-field. In the first theorem a formula is given for the absolute discriminant of \(L\). In the second theorem, it is shown that for any prime \(p\), \(L\) can be expressed as a compositum \(K_1K_2\dots K_n\) or \(K_0 K_1 K_2\dots K_n\), where each \(K_i\) is a cyclic \(q\)-field. Moreover, the second expression only occurs when \(q=p=2\). Here \(p\) is totally ramified in \(K_1\) or \(K_0\) and \(K_1\) whenever \(K_0\) is defined. Also, \(p\) is inert in \(K_2\) and splits completely in \(K_3\dots K_n\). On the basis of the latter theorem, the author classifies abelian \(q\)-fields into 4 or 8 types according to the decomposition of a fixed prime \(p\) in \(L\). Only 4 types occur unless \(q=p=2\). No proofs are given here, but the results would appear to follow readily by applying standard Hilbert theory to abelian extensions.
Mathematics Subject Classification ID: 11R20
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 1053789
zbMATH Keywords: abelian \(q\)-fields
zbMATH Keywords: absolute discriminant
author: Xianke Zhang
reviewed by: Charles J. Parry
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1053789