Field of Rational Functions is Field

Theorem
Let $K$ be a field, and let $K \left[{x}\right]$ be the integral domain of polynomial forms on $K$.

Let $K \left({x}\right)$ be the field of rational functions on $K$.

Then $K \left({x}\right)$ forms a field.

If the characteristic of $K$ is $p$, then the characteristic of $K \left({x}\right)$ is finite.

Comment
Thus we see that despite Characteristic of Finite Ring is Non-Zero (and by implication that of a finite field), it is not necessarily the case that the characteristic of an infinite field is zero.