Complex subfields of finite index

Question

It is known that the field $\mathbb{R}$ of real numbers is a complex subfield of index 2, that is, $[\mathbb{C},\mathbb{R}]=2$. Given an integer $n>2$ fixed, does there exist a subfield of $\mathbb{C}$ of index $n$?

Answer 1

There is no such subfield. It is a theorem of Artin-Schreier that if $K$ is algebraically closed and $L$ is a proper subfield of $K$ such that $[K:L]<\infty$, then $K$ is obtained from $L$ by adding a square root of $-1$, so $[K:L]=2$. See this MO answer.