Solution
= Solution
The relations are $\sigma^n=\tau^2=1$ and $\tau\sigma\tau=\sigma^{-1}$, so the generated <group> is $D_{2n}$. The <functions> $XY$ and $X^n+Y^n$ are fixed. Conversely $X^n,Y^n$ are roots of $T^2-(X^n+Y^n)T+(XY)^n$, and adjoining $X$ then $Y$ gives degree at most $2n$. Artin gives degree exactly $|G|=2n$, proving $M^G=\mathbb C(X^n+Y^n,XY)$.