Under the preceding quotient, $H/P_{\mathbb Q}(8\infty)$ is the subgroup
$$
\{1,7\}\leq(\mathbb Z/8\mathbb Z)^\times.
$$
It has order two in a group of order four. Hence $H$ contains $P_{\mathbb Q}(8\infty)$ and
$$
[I_{\mathbb Q}(8\infty):H]=2.
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 123 / 2 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-123/2/c/solution