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($g$). Show that:
(a) $F(x)$ is isomorphic to $F[x]\oplus F(x)_p $ as an $F$-vector space.
(b) If $\mathcal{I}=\{p(x)\in F[x] \mid p \text { is a monic irreducible polynomial}\}$, then
$$\beta= \{\frac{x^{j}}{p(x)^{k}} \mid p(x)\in \mathcal{I}, 0\leq j< \text{degree}(p); k\geq 1 \} $$ is a basis for $F(x)_p$ as an $F$-vector space.
I am done with part (a), but I don't know how to proceed in part (b). Can anyone help me please?
This is partial fractions.
$M=F(x)/F[x]$ is a torsion $F[x]$-module. Since $F[x]$ is a PID, $M$ is the direct sum of its $p(x)$-primary components, for irreducible polynomials $p(x) \in F[x]$. Show that any element in $F(x)/F[x]$ which is annihilated by a power of $p$ by can be represented in the form $r(x)/p(x)^n$, and use this to show that the elements you have written down form a basis of the $p(x)$-primary component of $F(x)/F[x]$.