Say $R$ is an integral domain and $k$ is a field.
Is it true that if $\bar{k}\supseteq R\supseteq k$, then $R$ is a field?
I'm not sure how to show this immediately, and it seems to be implied in the textbook, or else they are using some other facts not listed.
Yes, it is true. Let $\alpha$ be any element of $R$. Consider the homomorphism $\nu_\alpha:k[x]\to R$, $\nu_\alpha(p)=p(\alpha)$. Since $\alpha$ is algebraic over $k$, $\ker\nu_\alpha$ contains a non-zero polynomial. $\ker\nu_\alpha$ must therefore be a non-zero prime ideal of $k[x]$. Since $k[x]$ is PID, non-zero pime ideals are maximal. So $\operatorname{im}\nu_\alpha$ is a field: in other words, there is some $\beta\in\operatorname{im}\nu_\alpha$ such that $\beta\alpha=1$.