 Solution

The subgroup $H/P_{\mathbb{Q}}(8\infty )=\{1,7\}$ corresponds inside $\mathbb{Q}({\zeta }_8)$ to the fixed field of $\{1,c\}$, where $c$ is complex conjugation. This is the maximal real subfieldThus the existence theorem of global class field theory associates $H$ with $\boxed{\mathbb{Q}(\sqrt{2})}$.