TITLE: Does every infinite field contain a countably infinite subfield?
QUESTION [30 upvotes]: Does every infinite field contain a countably infinite subfield?
It's easy to see that every field $K$ contains either the rational numbers $\Bbb Q$ (when $K$ has characteristic $0$) or a finite field $\Bbb F_p$ (when $K$ has characteristic $p$). Thus, in the characteristic $0$ case, the answer is an easy "yes."
But if $K$ is infinite and has characteristic $p>0$, does the fact that $K \supset \Bbb F_p$ allow us to conclude that $K$ has a countably infinite subfield?

REPLY [6 votes]: Suppose $K$ has characteristic $p$.
If $K$ has a transcendental element $\alpha$, then $\mathbf{F}_p(\alpha) \subseteq K$ is countably infinite, being isomorphic to the rational function field $\mathbf{F}_p(x)$.
Otherwise, $K \subseteq \overline{\mathbf{F}}_p$ which is countable, so you can just take $K$ itself to be the countably infinite subfield.