Every ideal of $\mathbb Z$ prime to $8$ has a representative $n\mathbb Z$ with $n$ odd and positive. The <Artin reciprocity map> for $\mathbb Q(\zeta_8)/\mathbb Q$ is
$$
n\mathbb Z\longmapsto\sigma_n,\qquad
\sigma_n(\zeta_8)=\zeta_8^n.
$$
Its kernel consists exactly of positive principal ideals generated by numbers congruent to $1$ modulo $8$. Therefore
$$
I_{\mathbb Q}(8\infty)/P_{\mathbb Q}(8\infty)
\cong\operatorname{Gal}(\mathbb Q(\zeta_8)/\mathbb Q)
\cong(\mathbb Z/8\mathbb Z)^\times,
$$
and the four classes are
$$
\boxed{[\mathbb Z],[3\mathbb Z],[5\mathbb Z],[7\mathbb Z].}
$$
Complex conjugation sends $\zeta_8$ to $\zeta_8^{-1}=\zeta_8^7$, so it corresponds to $\boxed{7}$.

Solution

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

 Solution

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