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 $F$.

Let $K \left({x}\right)$ be the quotient field of $K[x]$.

Then $K(x)$ is the set of rational functions on $F$, i.e.:
 * $K \left({x}\right) = \left\{{\forall f \in K \left[{x}\right], g \in K \left[{x}\right]^*: \dfrac {f \left({x}\right)} {g \left({x}\right)}}\right\}$

where $K \left[{x}\right]^* = K \left[{x}\right] \setminus \left\{{\text{the null polynomial}}\right\}$.

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 although the characteristic of a 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.