Capitulation of $2$-class ideals of $protect mathbf{Q}(sqrt{-pq(2+sqrt{2})})$ where $pequiv qequiv pm 5;@mod ;8$

摘要

Let ${mathbf{K}}=mathbf{Q}(sqrt{-pq(2+sqrt{2})})$ where $p$ and $q$ are two different prime numbers such that $pequiv qequiv pm 5;@mod ;8$, ${mathbf{K}}_2^{(1)}$ the Hilbert $2$-class field of ${mathbf{K}}$, ${mathbf{K}}_2^{(2)}$ the Hilbert $2$-class field of ${mathbf{K}}_2^{(1)}$ and $G$ the Galois group of ${mathbf{K}}_2^{(2)}/{mathbf{K}}$. According to [4], $C_{2,{mathbf{K}}}$, the Sylow $2$-subgroup of the ideal class group of ${mathbf{K}}$ is isomorphic to ${mathbf{Z}}/{2{mathbf{Z}}}imes {mathbf{Z}}/{2{mathbf{Z}}}$, consequently ${mathbf{K}}_2^{(1)}/{mathbf{K}}$ contains three extensions ${mathbf{F}}_i/{mathbf{K}}$ $(i=1,2,3)$. In this paper, we are interested in the problem of capitulation of the classes of $C_{2,{mathbf{K}}}$ in ${mathbf{F}}_i$ $(i=1,2,3)$ and to determine the structure of $G$.