TITLE: Complex subfields of finite index
QUESTION [5 upvotes]: 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 $n$?

REPLY [5 votes]: 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.