# Field of Rational Functions is Field

## Theorem

Let $K$ be a field.

Let $K \sqbrk x$ be the integral domain of polynomial forms on $K$.

Let $\map K x$ be the field of rational functions on $K$.

Then $\map K x$ forms a field.

If the characteristic of $K$ is $p$, then the characteristic of $\map K x$ is non-zero.

## 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.