A explicit basis for the $\mathbb{C}$-vector space of rational function $\mathbb{C}(z)$.

Question

What would be a basis for this space? We know that for just the $\mathbb{C}[z]$ we have the basis to be $1, x, x^2, \ldots$. But when looking at the fraction field of $\mathbb{C}[x]$ does it still have the same basis? What about the monomials of the form $1/x^{n}$ for all $n$? Should they be considered in the basis as well? Instincts told me yes.

Answer 1

Hint: partial fraction decomposition.