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This article needs to be split, because there is a significant difference between rational functions in the function algebra of a field, vs what is described as “rational fractions” in this article, the field of fractions of a polynomial ring. The second are usually called “rational expressions” in typical school algebra; Frank Quinn calls the second object “polynomial fractions” in Proof Projects for Teachers. In particular, rational expressions form a field of fractions. However, rational functions over a field do not form a field of fractions, due to issues involving the definition of partial functions and division by zero.
Sounds good, please do split, if you have the energy.
The new definition seems to claim that $1/(x^p-x)$ in the field $\mathbf{Z}/p$ is not a rational function.
This contradicts the standard terminology: https://stacks.math.columbia.edu/tag/01RT. It also contradicts the Wikipedia article, which talks about the field of fractions of $k[X]$.
I suggest reverting to the old definition.
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