Capitulation of 2-class ideals of \(\mathbb Q\left(\sqrt{-pq(2+\sqrt 2)}\right)\) where \(p\equiv q\equiv {\pm 5} \bmod 8\)
type: Item
instance of: scholarly article
title: Capitulation of 2-class ideals of \(\mathbb Q\left(\sqrt{-pq(2+\sqrt 2)}\right)\) where \(p\equiv q\equiv {\pm 5} \bmod 8\)
zbMATH Open document ID: 1169.11046
author: Abdelmalek Azizi
author: Mohamed Talbi
published in: Annales Mathématiques Blaise Pascal
publication date: 2009-05-18T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10572
review text: This article concerns capitulation of ideals of the field \(K= \mathbb Q(\sqrt{-pq(2+\sqrt{2})})\) when \(p\) and \(q\) are distinct prime numbers with \(p\equiv q\equiv\pm 5\text{\,mod\,}9\). In particular, it discusses how many 2-ideal classes of \(K\) capitulate in the quadratic subextensions of \(K_2^{(1)}/K\), where \(K_2^{(1)}\) is the Hilbert 2-class field of \(K\). This is used to determine the isomorphism type of \(G=\text{Gal}(K^{(2)}_2/K)\), where \(K_2^{(2)}\) is the Hilbert 2-class field of \(K_2^{(1)})\). Along the way, systems of fundamental units for number fields related to \(K\) are found, in order that certain norm groups relevant to the issue of capitulation may be computed. Before summarizing the main results, we first observe that \(\text{Gal}(K_2^{(1)}/K)\) is the non-cyclic group of order 4 -- see \textit{E. Brown} and \textit{C. J. Parry} [ Pac. J. Math. 78, 11--26 (1978; Zbl 0405.12009)], so that \(K_2^{(1)}/K\) contains three proper subextensions \(F_1\), \(F_2\) and \(F_3\), all quadratic over \(K\). Théorème 3.2 states that when \(p\equiv q\equiv 3\bmod 8\), \(K^{(2)}_2= K^{(1)}_2\) so that \(G\) is also non-cyclic of order 4. Further, in this case all four 2-ideal classes of \(K\) capitulate in \(F_1\), \(F_2\) and \(F_3\). Capitulation of 2-ideal classes in \(F_1\), \(F_2\) and \(F_3\) when \(p\equiv q\equiv 5\bmod 8\) is less straightforward and depends upon the unit index of \(K_0= \mathbb Q(\sqrt{2}, \sqrt{pq})\), which is either 1 or 2. The content of Theorem 3.4 is to describe this capitulation in the two cases. Theorem 3.7 then deals with the isomorphism type of \(G\) when \(p\equiv q\equiv 5\bmod 8\), stating that \(G\) is dihedral when the afore-mentioned unit index is 1, and (generalized) quaternion when it is 2. In both cases, \(G\) has order \(2^m\) with \(m\geq 3\). The article concludes by demonstrating that the 2-class group of \(K(\sqrt{pq})\) in the case \(p\equiv q\equiv 5\bmod 8\) is cyclic and has order given in term…
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R37
zbMATH DE Number: 5554659
zbMATH Keywords: quartic fields
zbMATH Keywords: unit groups
zbMATH Keywords: Hilbert class field
zbMATH Keywords: capitulation
reviewed by: Paul Buckingham
MaRDI profile type: Publication
OpenAlex ID: W2275103875
cites work: Units of certain imaginary abelian number fields over \(\mathbb Q\).
cites work: Sur la capitulation des 2-classes d'idéaux de
cites work: Capitulation des 2-classes d'idéaux de Q(\sqrt 2,\sqrt d) où d est un entier naturel sans facteurs carrés
cites work: The 2-class group of biquadratic fields. II
cites work: Q4740641
cites work: Sur le 2-groupe des classes d'idéaux des corps quadratiques.
cites work: Number fields with class number congruent to 4 mod 8 and Hilbert's Theorem 94
cites work: Q5840258
cites work: Kuroda's class number formula
cites work: Q5541475
DOI: 10.5802/AMBP.253
description: scientific article; zbMATH DE number 5554659