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

For an unramified prime $\mathfrak{p}$ of $K$ and a prime $\mathfrak{P}$ of $L$ above it, the Frobenius automorphism is characterized byIn an abelian extension it is independent of $\mathfrak{P}$. This element is the Artin symbol

Every ideal of $\mathbb{Z}$ prime to $8$ has a representative $n\mathbb{Z}$ with $n$ odd and positive. The Artin reciprocity map for $\mathbb{Q}({\zeta }_8)/\mathbb{Q}$ isIts kernel consists exactly of positive principal ideals generated by numbers congruent to $1$ modulo $8$. Thereforeand the four classes areComplex conjugation sends ${\zeta }_8$ to ${\zeta }_8^{-1}={\zeta }_8^7$, so it corresponds to $\boxed{7}$.

Under the preceding quotient, $H/P_{\mathbb{Q}}(8\infty )$ is the subgroupIt has order two in a group of order four. Hence $H$ contains $P_{\mathbb{Q}}(8\infty )$ and

The ideal-theoretic existence theorem of global class field theory gives an inclusion-reversing correspondence between finite abelian extensions $L/K$ and congruence subgroupsThe corresponding field satisfies

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 $\boxed{\mathbb{Q}(\sqrt{2})}$.