= Maximal real subfield of a cyclotomic field
{title2=$\mathbb Q(\zeta_m)^+=\mathbb Q(\zeta_m+\zeta_m^{-1})$}
For $m\geq3$, put $\eta=\zeta_m+\zeta_m^{-1}$. Since $\eta$ is real and $\zeta_m$ is nonreal but satisfies $T^2-\eta T+1=0$, the <cyclotomic field> $\mathbb Q(\zeta_m)$ has degree two over $\mathbb Q(\eta)$. <Complex conjugation> fixes $\eta$ and is nontrivial. The <Fundamental theorem of Galois theory> therefore identifies its <fixed field> with $\mathbb Q(\eta)$. That <fixed field> is exactly $\mathbb Q(\zeta_m)\cap\mathbb R$, proving maximality among real <subfields>.
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