Fixed field

For a group $G$ of automorphisms of a field $L$, the fixed field isFor one automorphism $\sigma$, it is also written $L^{\sigma }$.

Automorphisms $\sigma (z)={\zeta }_{n}z$ 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.

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.