TITLE: Is there a field extension over the real numbers that is not the same as the field of complex numbers?
QUESTION [8 upvotes]: I'm trying to determine if there is a field, F, such that $\mathbb{R}$ $\subsetneq$ F $\subsetneq$ $\mathbb{C}$ where F is not the same as $\mathbb{R}$ or $\mathbb{C}$.

REPLY [5 votes]: Suppose $F$ is a field such that $\mathbb{R} \subsetneq F \subseteq \mathbb{C}.$ Then $F$ contains at least one element $a+bi$ where $b\neq 0.$ Since $F$ contains all the reals, it contains $b^{-1}$ and $-a,$ so it contains $b^{-1}\left((a+bi)-a\right)=i.$ Hence it contains all complex numbers and $F=\mathbb{C}.$