The Capitulation Problem for certain Number Fields

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.