On computing discriminants of subfields of \(\mathbb Q(\zeta_{p^r})\)
type: Item
instance of: scholarly article
title: On computing discriminants of subfields of \(\mathbb Q(\zeta_{p^r})\)
zbMATH Open document ID: 1031.11065
author: Trajano Pires da Nóbrega Neto
author: J. Carmelo Interlando
author: José Othon Dantas Lopes
published in: Journal of Number Theory
publication date: 2003-03-23T00:00:00Z
review text: Let \(p\) be an odd prime, \(r\) a positive integer and \(K\) a subfield of the \(p^r\)th cyclotomic field \(\mathbb Q(\zeta_{p^r})\). Then the absolute degree of \(K\) is of the form \(up^j\), where \(u\) is a divisor of \(p-1\) and \(j\) is an integer satisfying \(0\leq j\leq r-1\). Due to the well-known cyclicity of the Galois group of \(\mathbb Q(\zeta_{p^r})/\mathbb Q\), its absolute degree fully characterizes \(K\) among the subfields of \(\mathbb Q(\zeta_{p^r})\). The main result of the paper consists in the computation of the discriminant of \(K\) via Hasse conductor-discriminant formula. Is is shown that the absolute value of the discriminant of \(K\) is equal to \(p^a\), where \(a=u((j+2)p^j-(p^{j+1}-1)/(p-1))-1\).
reviewed by: Radan Kučera
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R18
Mathematics Subject Classification ID: 11R20
zbMATH DE Number: 1886721
zbMATH Keywords: Dirichlet characters
zbMATH Keywords: conductors
zbMATH Keywords: cyclotomic fields
zbMATH Keywords: discriminants of abelian number fields
zbMATH Keywords: Hasse conductor-discriminant formula
MaRDI profile type: Publication
full work available at URL: jnth.2002.2796
OpenAlex ID: W1483105767
cites work: Q5521595
cites work: Q4317890
cites work: Q3943047
DOI: 10.1006/JNTH.2002.2796
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description: scientific article; zbMATH DE number 1886721