= 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, and
$$
K^{\langle\sigma,\tau\rangle}
=\mathbb Q(x^2+y^2,x^2y^2).
$$
The 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.
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