Past exam of the galois theory course of the University of Oxford / 2023 / 1 / b

The element $\zeta$ satisfies $T^2-\eta T+1=0$, so $[\mathbb{Q}(\zeta ):\mathbb{Q}(\eta )]\le 2$. Since $\eta =\zeta +\overline{\zeta }$ is real, $\mathbb{Q}(\eta )\subseteq \mathbb{R}$. A primitive root of order $m\ge 3$ is nonreal: the only real roots of unity are $\pm 1$, of orders at most two. Hence $\zeta \notin \mathbb{Q}(\eta )$ andEquivalently, nontrivial complex conjugation fixes precisely the maximal real subfield of a cyclotomic field in this degree-two extension.