$\mathbf {The \ Problem \ is}:$ Let, $f \in \mathbb Q[x]$ be a polynomial of degree $n \gt 0$ and let $p_1,p_2, \cdots p_{n+1}$ be distinct prime numbers . Show that there exists a non-zero polynomial $g \in \mathbb Q[x]$ such that $fg = \sum_{i=1}^{n+1} c_iX^{p_i}$ with $c_i \in \mathbb Q .$
$\mathbf {My \ approach} :$ Actually, I was thinking to approach by induction with the fact that $g \in \langle x\rangle$ as the rt hand side has no constant term. And, if we consider $f$ to be monic, then $f=x^{n+1}+f_1$ where $deg(f_1) \lt n$, then applying the induction hypothesis, we get a $g$, but then by comparison of co-efficients on both sides, the process becomes lengthy.
Again, $\mathbb Q[x]$ is a principal ideal domain , and for any distinct set of primes $p_1,p_2, \cdots p_{n+1}$, consider the smallest ideal containing $\{X^{p_i} | i=1,2 ...,n+1\}$ and name that ideal by $I_{n+1}$, then $I_{n+1}$ is also principal, but I can't approach using these paths .
A small hint is warmly appreciated .