Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 4 / 18I (Galois Theory)

An action by field automorphisms is a homomorphism $Go\mathrm{Aut}(L)$. Elements fixed by every group element are closed under addition, multiplication, additive inverses, and inversion of nonzero elements, so $L^G$ is a subfield.

The orbit polynomial ${\prod }_{\beta \in G\alpha }(X-\beta )$ has coefficients fixed by $G$, lies in $K[X]$, and has distinct roots. Thus $\alpha$ is separable and its degree is the orbit size $|G:G_{\alpha }|$, which divides $|G|$. If $L=K(\alpha )$ and the action is faithful, only the identity fixes $\alpha$, so the degree is $|G|$.

The relations ${\sigma }^n={\tau }^2=1$ and $\tau \sigma \tau ={\sigma }^{-1}$ give the dihedral group; its $2n$ substitutions $z\mapsto {\zeta }_n^{j}z$ and $z\mapsto {\zeta }_n^j/z$ are distinct. The element $w=z^n+z^{-n}$ is fixed. Since $z$ satisfies $X^{2n}-w{X}^n+1=0$, the extension has degree at most $2n$, while the automorphism group gives degree at least $2n$. Hence $L^G=\mathbb{C}(w)$.