Does there exist a subfield $S$ of $\mathbb C$ such that $\mathbb R \subset S \subset \mathbb C$?

Question

Does there exist a subfield $S$ of $\mathbb C$ such that $\mathbb R \subset S \subset \mathbb C$ ? ; I kind of have a feeling that there does not exist any such $S$ but cannot prove . Thanks in advance for any help.

Answer 1

Suppose $S$ is a field, $\mathbb R\subsetneq S\subseteq\mathbb C$. Choose $z\in S\setminus\mathbb R$. Then $z=a+bi$ where $a,b\in\mathbb R,\,b\ne0$, and $i=\frac{z-a}b\in S$, whence $S=\mathbb C$.

Answer 2

If $S$ is a field such that $\mathbb R\subseteq S\subseteq \mathbb C$, then $S$ is a vector space over $\mathbb R$ which is a subspace of a vector space of dimension $2$, $\mathbb C$. This means that $S=\mathbb R$ or $S=\mathbb C$.