Dihedral fixed field in one rational variable ($\mathbb{C}(z{)}^{D_{2n}}=\mathbb{C}(z^n+z^{-n})$)

Automorphisms $\sigma (z)={\zeta }_{n}z$ and $\tau (z)=1/z$ generate a dihedral group of order $2n$. The invariant $w=z^n+z^{-n}$ has rational map degree $2n$, so $[\mathbb{C}(z):\mathbb{C}(w)]=2n$. The fixed-field theorem for a finite automorphism group gives the same degree over its fixed field, proving equality. For odd prime $n$, the nontrivial proper subgroups are the rotation group and the $n$ reflection groups. Their fixed fields are $\mathbb{C}(z^n)$ and $\mathbb{C}(z+{\zeta }_n^{-j}/z)$ for reflections ${\sigma }^j\tau$, respectively.