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Can anyone give an explicit basis of the $k$-vector space $k(X) = \operatorname{Quot}(k[X])$ of rational functions over $k$?

The dimension is given by $$\dim_k k(X) = \max(|k|, |\mathbb N|).$$

If $k$ is infinite, this follows from $|k| \leqslant |k(X)| \leqslant |k[X] \times k[X]| = |k \times k| = |k|$ and the linear independence of $\lbrace \frac{1}{X - \alpha} \mid \alpha \in k\rbrace.$ If $k$ is finite, the result can be obtained similarly from $|k(X)|=|\mathbb N|$ and the linear independence of the monomials $X^n$, $n \geqslant 0$.

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    One way to do this is using partial fractions decompositions.2012-04-01

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