<Artin fixed-field theorem> says that for a finite <group> $G\le\operatorname{Aut}L$, $L/L^G$ is Galois with <group> $G$ and degree $|G|$. Let any finite $G$ act faithfully by left translation on variables $X_g$ in $L=\mathbb C(X_g:g\in G)$. Artin's theorem then realizes $G$ as $\operatorname{Gal}(L/L^G)$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 18F / a / Solution

 Solution (source code)