Capitulation des 2-classes d'idéaux de $\mathbf{k}=\mathbb{Q}(\sqrt{2p},i)$

Abstract

Let $p$ be a prime number such that $p\equiv 1$ mod $8$ and $i=\sqrt{-1}$. Let $\mathbf{k}=\mathbb{Q}(\sqrt{2p},i)$, ${\mathbf{k}}_1^{(2)}$ be the Hilbert $2$-class field of $\mathbf{k}$, ${\mathbf{k}}_2^{(2)}$ be the Hilbert $2$-class field of ${\mathbf{k}}_1^{(2)}$ and $G=\mathrm{Gal}({\mathbf{k}}_2^{(2)}/\mathbf{k})$ be the Galois group of ${\mathbf{k}}_2^{(2)}/\mathbf{k}$. Suppose that the $2$-part, $C_{\mathbf{k},2}$, of the class group of $\mathbf{k}$ is of type $(2,4)$; then ${\mathbf{k}}_1^{(2)}$ contains six extensions ${\mathbf{K}}_{\mathbf{i},\mathbf{j}}/\mathbf{k}$, $i=1,2,3$ and $j=2,4$. Our goal is to study the problem of the capitulation of $2$-ideal classes of ${\mathbf{K}}_{\mathbf{i},\mathbf{j}}$ and to determine the structure of $G$.

Cited by in corpus (2)

• On the rank of the $2$-class group of some imaginary triquadratic number fields
• On the Hilbert $2$-class field towers of some cyclotomic ${\mathbb{Z}}_2$-extensions