Field of Rational Functions
Contents |
Theorem
Let $K$ be a field, and let $K \left[{x}\right]$ be the integral domain of polynomial forms on $K$.
Let $K \left({x}\right)$ be the set of rational functions on $K$, i.e.:
- $K \left({x}\right) = \left\{{\forall f \in K \left[{x}\right], g \in K \left[{x}\right]^*: \dfrac {f \left({x}\right)} {g \left({x}\right)}}\right\}$
where $K \left[{x}\right]^* = K \left[{x}\right] \setminus \left\{{\text{the null polynomial}}\right\}$.
Then $K \left({x}\right)$ forms a field.
If the characteristic of $K$ is $p$, then the characteristic of $K \left({x}\right)$ is finite.
Proof
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Comment
Thus we see that although the characteristic of a 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
- C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 4.17$: Example $26$
