Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 2 / 18F / c / Solution

Identify $G\simeq (\mathbb{Z}/15\mathbb{Z}{)}^{\times }\simeq C_4\times C_2$, with $\sigma (\zeta )={\zeta }^2$ and $\tau (\zeta )={\zeta }^{-1}$. Its subgroups areTheir fixed fields, in the same layers, areThe real fields are $\mathbb{Q}$, $\mathbb{Q}(\sqrt{5})$, and $L^{+}$. This is the Galois correspondence.