= Solution
The automorphisms satisfy $\sigma^n=\tau^2=1$ and $\tau\sigma\tau=\sigma^{-1}$. The $n$ rotations $z\mapsto\zeta_n^jz$ and $n$ inversion maps $z\mapsto\zeta_n^j/z$ are all distinct, so the <dihedral group> $G$ has order $2n$, including the case $n=1$ when it has order two. The invariant
$$
\boxed{w=z^n+z^{-n}}
$$
is fixed by both generators.
Use two standard field-theoretic results: the fixed-field theorem for a finite group of automorphisms gives $[L:L^G]=|G|$ and makes $L/L^G$ Galois with group $G$; and a nonconstant rational function $P(z)/Q(z)$ in lowest terms satisfies $[\mathbb C(z):\mathbb C(P/Q)]=\max(\deg P,\deg Q)$. Here $w=(z^{2n}+1)/z^n$ has coprime numerator and denominator and degree $2n$. Thus $[L:\mathbb C(w)]=[L:L^G]=2n$. Since $\mathbb C(w)\subseteq L^G$, the tower law proves the <dihedral fixed field in one rational variable>:
$$
\boxed{L^G=\mathbb C(z^n+z^{-n}).}
$$
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