On the class group of an imaginary cyclic field of conductor \(8p\) and \(2\)-power degree
type: Item
instance of: scholarly article
title: On the class group of an imaginary cyclic field of conductor \(8p\) and \(2\)-power degree
zbMATH Open document ID: 1480.11141
author: Humio Ichimura
author: Hiroki Sumida
published in: Tokyo Journal of Mathematics
publication date: 2021-09-13T00:00:00Z
review text: Let \(p=2^{e+1}q+1\) (with \(q\) odd) be a prime and denote by \(K\) the imaginary cyclic subfield of \(\mathbb Q(\zeta_p)\) of degree \(2^{e+1}\). Let \(F\ne K\) be the imaginary subfield of \(K(\sqrt 2)\) containing \(K^+\), the maximal real subfield of \(K\). The authors determine the structure of the \(2\)-class group \(A_F\) of \(F\). Let \(\kappa\) the smallest integer such that the prime \(p\) splits completely in \(\mathbb Q(2^{e-\kappa+1})\). It is shown in Theorem \(1\) that if \(e\ge2\) and \(\kappa\ge1\), then \(A_F\) is the direct sum of \(2^{e-\kappa+1}\) copies of \(\mathbb Z/2\mathbb Z\). The case \(e\ge2\) and \(\kappa=0\) is more complicated and is treated in Theorem 2.
Mathematics Subject Classification ID: 11R20
Mathematics Subject Classification ID: 11R29
zbMATH DE Number: 7394485
zbMATH Keywords: imaginary cyclic feld
zbMATH Keywords: class-group
reviewed by: Władysław Narkiewicz
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1502179326
OpenAlex ID: W3120416639
cites work: On the Existence of Fields Governing the 2-Invariants of the Classgroup of Q(√dp) as p Varies
cites work: Q3994474
cites work: Q3820703
cites work: Q5421238
cites work: Lectures on P-Adic L-Functions. (AM-74)
cites work: 8-RANKS OF CLASS GROUPS OF IMAGINARY QUADRATIC NUMBER FIELDS AND THEIR DENSITIES
cites work: Cyclotomic Z//2-extensions of J-fields
cites work: On the 16-rank of class groups of \({\mathbb{Q}(\sqrt{-8p})}\) for \({p \equiv -1 \bmod 4}\)
cites work: The quadratic number fields with cyclic 2-classgroups
cites work: Q4821147
cites work: Q4753369
cites work: Arithmetischer Beweis des Satzes über die Anzahl der durch vier teilbaren Invarianten der absoluten Klassengruppe im quadratischen Zahlkörper.
cites work: Q5690489
cites work: The Iwasawa conjecture for totally real fields
cites work: Divisibility by 16 of class number of quadratic fields whose 2-class groups are cyclic
cites work: On the Class Number of a Relatively Cyclic Number Field
cites work: The generalized Rédei-matrix
cites work: Class groups under relative quadratic extensions
cites work: ANOTHER CASE OF A SCHOLZ'S THEOREM ON CLASS GROUPS
DOI: 10.3836/TJM/1502179326
Recommended article: On the class groups of certain imaginary cyclic fields of 2-power degree
Recommended article: Q5887379
Recommended article: 8-ranks of class groups of some imaginary quadratic number fields
Recommended article: Imaginary cyclic fields of degree p-1 whose ideal class groups have p-rank at least two
Recommended article: Determination of All Nonquadratic Imaginary Cyclic Number Fields of 2-Power Degrees with Ideal Class Groups of Exponents ≤2
Recommended article: The nonquadratic imaginary cyclic fields of 2-power degrees with class numbers equal to their genus class numbers
Recommended article: 8-RANKS OF CLASS GROUPS OF IMAGINARY QUADRATIC NUMBER FIELDS AND THEIR DENSITIES
Recommended article: Q3798753
Recommended article: The 2-Rank of the Class Group of Imaginary Bicyclic Biquadratic Fields
Recommended article: On the class numbers of real cyclotomic fields of conductor pq
description: scientific article; zbMATH DE number 7394485