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VOL. 30 | 2001 The Capitulation Problem for certain Number Fields
Mohammed Ayadi, Abdelmalek Azizi, Moulay Chrif Ismaili

Editor(s) Katsuya Miyake

Abstract

We study the capitulation problem for certain number fields of degree 3, 4, and 6.

(I) Capitulation of the 2-ideal classes of $\mathbb{Q}(\sqrt{d}, i)$ (by A. AZIZI)

Let $d \in \mathbb{N}$, $i = \sqrt{-1}$, $\mathbf{k} = \mathbb{Q}(\sqrt{d}, i)$, ${\mathbf{k}}_1^{(2)}$ be the Hilbert 2-class field of $\mathbf{k}$, ${\mathbf{k}}_2^{(2)}$ be the Hilbert 2-class field of ${\mathbf{k}}_1^{(2)}$, $C_{\mathbf{k},2}$ be the 2-component of the ideal class group of $\mathbf{k}$ and $G_2$ the Galois group of ${\mathbf{k}}_2^{(2)}/\mathbf{k}$. We suppose that $C_{\mathbf{k},2}$ is of type (2, 2); then ${\mathbf{k}}_1^{(2)}$ contains three extensions $F_i/\mathbf{k}$, $i = 1, 2, 3$. The aim of this section is to study the capitulation of the 2-ideal classes in $F_i$, $i = 1, 2, 3$, and to determine the structure of $G_2$.

(II) On the capitulation of the 3-ideal classes of a cubic cyclic field (by M. AYADI)

Let $k$ be a cubic cyclic field over $\mathbb{Q}$, and ${\mathbf{k}}_1^{(3)}$ the Hilbert 3-class field of $\mathbf{k}$. If the class number of $\mathbf{k}$ is exactly divisible by 9, then its 3-ideal class group is of type (3, 3), and ${\mathbf{k}}_1^{(3)}$ contains four cubic extensions ${\mathbf{K}}_i/\mathbf{k}$ in which we study the capitulation problem for the 3-ideal classes of $\mathbf{k}$.

(III) On the capitulation of the 3-ideal classes of the normal closure of a pure cubic field (by M. C. ISMAILI)

Let $\Gamma = \mathbb{Q}(\sqrt[3]{n})$ be a pure cubic field, $\mathbf{k} = \mathbb{Q}(\sqrt[3]{n}, j)$ its normal closure ($j = e^{\frac{2i\pi}{3}}$), ${\mathbf{k}}_1^{(3)}$ the Hilbert 3-class field of $\mathbf{k}$, and let $S_{\mathbf{k}}$ be the 3-ideal class group of $\mathbf{k}$. When $S_{\mathbf{k}}$ is of type (3, 3), we study the capitulation of the 3-ideal classes of $S_{\mathbf{k}}$ in the four intermediate extensions of ${\mathbf{k}}_1^{(3)}/\mathbf{k}$, and we show that if the class number of $\Gamma$ is divisible by 9, then we have some necessary conditions on $n$. We have also some informations about the unit group of $\mathbf{k}$ in some cases.

Information

Published: 1 January 2001
First available in Project Euclid: 13 September 2018

zbMATH: 1015.11055
MathSciNet: MR1846473

Digital Object Identifier: 10.2969/aspm/03010467

Rights: Copyright © 2001 Mathematical Society of Japan

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