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.

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 $26$