If $\zeta$ is a primitive nth root of unity, it appears that the cyclotomic field $\mathbb{Q}(\zeta )$can be partitioned into ${\mathbb{Q}}(\zeta +{{\zeta }^{-1}})$ and ${\mathbb{Q}}(\zeta -{{\zeta }^{-1}})$ where ${{\mathbb{Q}}^{+}}$= ${\mathbb{Q}}(\zeta +{{\zeta }^{-1}})$is the well-known maximal totally real subfield of $\mathbb{Q}(\zeta )$. This subfield has dimension $\varphi (n)/2$ over $\mathbb{Q}$, so ${{\mathbb{Q}}^{-}}$= ${\mathbb{Q}}(\zeta -{{\zeta }^{-1}})$ should also have dimension $\varphi (n)/2$. Some authors refer to ${{\mathbb{Q}}^{-}}$as the maximal totally complex subfield of $\mathbb{Q}(\zeta )$but I have never seen a proof of this.
Since ${{\mathbb{Q}}^{+}}$= $\mathbb{Q}(\zeta )$$\cap \mathbb{R}$, the complement as a vector space is $\mathbb{Q}(\zeta )\cap \mathbb{R}i$ and each of these must have dimension $\varphi (n)/2$. ${{\mathbb{Q}}^{+}}$ is always ${\mathbb{Q}}(\zeta +{{\zeta }^{-1}})$, and the question is how to characterize $\mathbb{Q}(\zeta )\cap \mathbb{R}i$. The symmetric ‘odd’ representation would be ${\mathbb{Q}}(\zeta -{{\zeta }^{-1}})$but clearly that does not always hold.
