# Field of Rational Functions is Field

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## 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.

## Proof

## 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.

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $4$: Fields: $\S 17$. The Characteristic of a Field: Example $26$