Restricted Multiples

We show that polynomials admit multiples supported on prime exponents.

Proposition 1. Let \(f\) be a polynomial over a field \(k\). Then there is a polynomial \(g\) divisible by \(f\) which is a linear combination of the polynomials \[\{x^p:p\text{ is prime}\}.\]

Proof. Observe that \[\frac{k[x]}{(f(x))}\] is a finite-dimensional vector space. Thus, some nonzero linear combination of the polynomials in \[\{x^p:p\text{ is prime}\}.\] is required to vanish in the quotient \(k[x]/(f(x))\). The result follows. \(\blacksquare\)

Remark 2. Of course, there is noting special about primes, as the proof makes clear.