Dihedral fixed field of a two-variable rational function field

On $K=\mathbb{Q}(x,y)$, let $\sigma (x,y)=(y,-x)$ and $\tau (x,y)=(x,-y)$. They generate a dihedral group of order eight, andThe inclusion from right to left is immediate. In the other direction, $x^2,y^2$ solve a quadratic over the displayed field and adjoining their square roots has degree at most four; the resulting total degree is at most eight, while the automorphism-count bound for a finite field extension gives the reverse bound.