Is it possible for an infinite field that does not contain a subfield isomorphic to $\Bbb Q$?

Question

If $F$ is a field, then $\text{char}F=0$ or $p$, a prime. If $\text{char}F=0$, we know that it contains a subfield which is isomorphic to $\Bbb Q$. However, there exists field that is infinite and its char. is $p$. So it induce the following question : is it possible for an infinite field that does not contain a subfield isomorphic to $\Bbb Q$?

Answer 1

Sure. No infinite field with characteristice $p$ has a subfield isomorphic to $\mathbb Q$. Take, for instance $\mathbb{F}_p(x)$, for any prime $p$.

Answer 2

Another example of an infinite field with characteristic $p>0$ is $$\mathbb{F}_p^a=\bigcup_{n \in \mathbb N} \mathbb{F}_{p^n}$$ the algebraic closure of $\mathbb F_p$.

See article INFINITE RINGS AND FIELDS WITH POSITIVE CHARACTERISTIC for more details.