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

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 $\boxed{7}$.