Let $\alpha\in\mathbb{C}\setminus\mathbb{R}$ be algebraic with $\alpha=a+bi$, $a,b\in\mathbb{R}$ and the field extension $\mathbb{Q}(\alpha)|\mathbb{Q}$, with $[\mathbb{Q}(\alpha):\mathbb{Q}]=n$.
My question is: is there any way to establish a relationship between the degree of the extensions $\mathbb{Q}(a)|\mathbb{Q}$, $\mathbb{Q}(bi)|\mathbb{Q}$ and $n$?
I tried to use that both $\alpha$ and $\overline{\alpha}$ are roots of the irreducible polynomial of $\alpha$ and that $a=\frac{1}{2}(\alpha+\overline{\alpha})$ and $bi=\frac{1}{2}(\alpha-\overline{\alpha})$ with no success.