Let $p(X)\in F[X]$ and $W$ a subspace of $F[X]$, show that $p(X)W=W\Rightarrow p(X)$ is a constant polynomial

Question

Let $F$ be a field and let $W$ be a nontrivial subspace of the vector space $F[X]$ over $F$. Let $p(X) ∈ F[X]$ be a given monic polynomial and let $p(X)W = \{p(X)f(X) \mid f(X) ∈ W\}$. Show that $p(X)W$ is a subspace of $F[X]$ and find a necessary and sufficient condition for it to equal $W$.

The first part is trivial since $F[X]$ is an associative $F$-algebra with polynomial product, and I think the necessary and sufficient condition is that $p(X)$ is a unit (constant nonzero polynomial), so I need help in how to show that if $p(X)W=W$, then $p(X)$ has inverse or is a constant polynomial.

This is an exercise from the book: The Linear Algebra a Beginning Graduate Student Should Know by Golan.

Answer 1

Let $f(X)\in W$, $f\ne0$. In order for $f\in p(X)W$ you need to find $g(X)\in W$ such that $p(X)g(X)=f(X)$.

Since you can take $f$ of minimal degree…