Solution
= Solution
The relations $σ^n=τ^2=1$ and $τσ τ=σ^{-1}$ give the dihedral group; its $2n$ substitutions $z\mapstoζ_n^jz$ and $z\mapstoζ_n^j/z$ are distinct. The element $w=z^n+z^{-n}$ is fixed. Since $z$ satisfies $X^{2n}-wX^n+1=0$, the extension has degree at most $2n$, while the automorphism group gives degree at least $2n$. Hence $L^G=\mathbb C(w)$.