Suppose $F$ is a field, and let $F(x)$ denote the $F$-vector space of all rational functions $\frac{f(x)}{g(x)}$, where $f,g\in F[x]$ are polynomials, with $g$ different from zero. Let $F(x)_p$ denote the subspace of F(x) of all proper fractions, i.e. all $\frac{f(x)}{g(x)}$ where degree($f$) $<$ degree($g$). It is know that
$$\beta= \{\frac{x^{j}}{p(x)^{k}} \mid p(x) \text{is a monic irreducible polynomial}, 0\leq j< \text{degree}(p); k\geq 1 \}$$
is a basis for $F(x)_p$ as an $F$-vector space. What other basis that could be constructed for F(x)? I mean is there any other basis for F(x)?
Can anyone help me please? or provide me a reference about basis specially if F is finite.