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