= Solution
The element $\zeta$ satisfies $T^2-\eta T+1=0$, so $[\mathbb Q(\zeta):\mathbb Q(\eta)]\leq2$. Since $\eta=\zeta+\overline\zeta$ is real, $\mathbb Q(\eta)\subseteq\mathbb R$. A primitive root of order $m\geq3$ is nonreal: the only real <roots of unity> are $\pm1$, of orders at most two. Hence $\zeta\notin\mathbb Q(\eta)$ and
$$
\boxed{[\mathbb Q(\zeta):\mathbb Q(\eta)]=2}.
$$
Equivalently, nontrivial <complex conjugation> fixes precisely the <maximal real subfield of a cyclotomic field> in this degree-two extension.
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