What is the maximal real subfield in a cyclotomic extension \(\mathbb{Q}(\zeta_n)/\mathbb{Q}\)?
Solution
Let \[ K=\mathbb Q(\zeta_n). \] Complex conjugation acts by \[ \zeta_n\longmapsto\zeta_n^{-1}. \] Therefore the maximal real subfield is the fixed field \[ K^+=K^{\langle c\rangle}=K\cap\mathbb R. \] The element \[ \zeta_n+\zeta_n^{-1}=2\cos\frac{2\pi}{n} \] is fixed by conjugation, and \(\zeta_n\) satisfies \[ x^2-(\zeta_n+\zeta_n^{-1})x+1=0 \] over \(\mathbb Q(\zeta_n+\zeta_n^{-1})\). For \(n>2\), \(K\) is not real, so \[ [K:K^+]=2. \] Hence \[ \boxed{K^+=\mathbb Q(\zeta_n+\zeta_n^{-1})} \] and \[ [K^+:\mathbb Q]=\frac{\varphi(n)}2. \] Since \[ \operatorname{Gal}(K/\mathbb Q)\cong(\mathbb Z/n\mathbb Z)^\times \] and complex conjugation corresponds to \(-1\), for \(n>2\), \[ \operatorname{Gal}(K^+/\mathbb Q) \cong(\mathbb Z/n\mathbb Z)^\times/\{\pm1\}. \] For \(n=1,2\), the cyclotomic field is already \(\mathbb Q\).