Let $\mathbb{F}$ a field such that $\mathbb{R}\subset\mathbb{F}\subset\mathbb{C}$. Prove that $\mathbb{F}=\mathbb{R}$ or $\mathbb{F}=\mathbb{C}$.
What I know is, that if $\mathbb{F}\neq\mathbb{R}$ it means that exists $a\in\mathbb{F}$ such that $a\in\mathbb{C}$ and $a\notin\mathbb{R}$. But I don't know how to use it. since I don't know the actions of $\mathbb{F}$ can't it just be defined with the special case of $a$ in mind and still hold all the axioms?
Also, this is from homework in Linear Aglebra (1), where the rest of the questions are about bases of vector spaces and dimensions and such, but there are no vector spaces here so I'm not really sure how it fits.