Sylow subgroups of ideal class groups with moduli
type: Item
instance of: scholarly article
title: Sylow subgroups of ideal class groups with moduli
zbMATH Open document ID: 0723.11053
DOI: 10.1016/0022-314X(90)90096-A
author: Ajchara Harnchoowong
author: Wen-Ching Winnie Li
published in: Journal of Number Theory
publication date: 1990-01-01T00:00:00Z
review text: Let K/k be a cyclic extension of number fields of prime degree \(\ell\). The authors generalize prior work on the Galois structure of the \(\ell\)- Sylow subgroup of the ideal class group of K, especially results of \textit{G. Gras} [Ann. Inst. Fourier 23, No.3, 1-48 (1973; Zbl 0276.12013); No.4, 1-44 (1974; Zbl 0262.12007)] and \textit{P. Morton} [J. Reine Angew. Math. 307/308, 373-398 (1979; Zbl 0395.12018)]. Here the authors consider the \(\ell\)-Sylow subgroup L of the ideal class group of K modulo \(f\infty\), where f is a nonzero integral ideal of k and \(\infty\) is a (possibly trivial) product of real places of k. Results on the structure of L are obtained under the assumption that the ideal f is prime to both \(\ell\) and the discriminant of K/k. For this purpose, a filtration of L of the type first considered by \textit{E. Inaba} [J. Fac. Sci. Univ. Tokyo, Sect. I 4, 61-115 (1940; Zbl 0024.01002)] is used. Let \(\sigma\) be a generator of the Galois group of K/k and define \(L_ i=\{x\in L:\;x\in \ker (\sigma -1)^ i\}.\) The authors calculate the orders of \(L_ 1\) and the successive quotients in the resulting filtration \(L_ 1\subseteq L_ 2\subseteq...\subseteq L\) of subgroups of L, and give an algorithm yielding the generators of the subgroups occurring there. Under an additional assumption on the triviality of norms of elements of L, the \(\ell^ r\)-rank of L for \(r\geq 1\) is expressed as a product of the orders of certain of the quotients.
reviewed by: Andrew G. Earnest
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 11R20
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 11R33
zbMATH DE Number: 4191813
zbMATH Keywords: Galois module structure
zbMATH Keywords: cyclic extension
zbMATH Keywords: \(\ell\)-Sylow subgroup
zbMATH Keywords: ideal class group
zbMATH Keywords: filtration
MaRDI profile type: Publication
cites work: Zur Berechnung der 2-Klassenzahl der quadratischen Zahlkörper mit genau zwei verschiedenen Diskrimantenprimteilern.
cites work: Die 2-Klassenzahlen spezieller quadratischer Zahlkörper.
cites work: Q5532660
cites work: Sur les \(\ell\)-classes d'idéaux dans les extensions cycliques rélatives de degré premier \(\ell\). II
cites work: Über die Klassenzahl des Körpers \(\text{P}\left(\sqrt{-p}\right)\) mit einer Primzahl \(p\equiv1\pmod{2^ 3}\)
cites work: Über die Klassenzahl des Körpers \(P(\sqrt{-2p})\) mit einer Primzahl \(p\ne 2\)
cites work: Über die Teilbarkeit durch 23 der Klassenzahl imaginär-quadratischer Zahlkörper mit genau zwei verschiedenen Diskriminantenprimteilern.
cites work: Über die Teilbarkeit durch 23 der Klassenzahl der quadratischen Zahlkörper mit genau zwei verschiedenen Diskriminantenprimteilern
cites work: Q4109740
cites work: On the Computational Complexity of Determining the Solvability or Unsolvability of the Equation X 2 - DY 2 = -1
cites work: On Rédei's theory of the Pell equation.
cites work: Arithmetischer Beweis des Satzes über die Anzahl der durch vier teilbaren Invarianten der absoluten Klassengruppe im quadratischen Zahlkörper.
cites work: Die 2-Ringklassengruppe des quadratischen Zahlkörpers und die Theorie der Pellschen Gleichung
cites work: Q4753369
cites work: Gauss's Ternary Form Reduction and the 2-Sylow Subgroup
cites work: Pieces of eight in class groups of quadratic fields
description: scientific article; zbMATH DE number 4191813