= Dihedral fixed field in one rational variable
{title2=$\mathbb C(z)^{D_{2n}}=\mathbb C(z^n+z^{-n})$}
Automorphisms $\sigma(z)=\zeta_nz$ 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.
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