# Field of Rational Functions is Field

(Redirected from Field of Rational Fractions is Field)

Jump to navigation
Jump to search
## 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

This theorem requires a proof.In particular: Reverted this to the version 9th May 2011 as it's mutated into something that makes less sense.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## 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 $26$