Question

Prove that rational numbers is a smallest infinite field?

Answer 1

Solution:

Proof:

Suppose K has characteristic $b$ .

If K has a transcendental element $a$ , then $\mathbf{F}_{b}(a) \subseteq K$ is countably infinite, being isomorphic to the rational function field $\mathbf{F}_{p}(x)$ .

Otherwise, $K \subseteq \overline{\mathbf{F}}_{b}$ which is countable, so it can be just taken K itself to be the countably infinite subfield. Thus, the rational numbers is a smallest infinite field