Let $K$ be a subfield/ring of a field/ring $F$. Is there a nontrivial example of the product of a polynomial over $K$ and a polynomial not over $K$ giving a polynomial over $K$?
For a simple version is there a polynomial with integer coefficients that multiplied buy a polynomial with some real coefficients gives integer coefficient?
I'm guessing the case that $K$ is a field is more difficult because we have inverses unlike $\Bbb Z.$ Is there a counterexample with $K=\Bbb Q$ and $F=\Bbb R$ or $\Bbb C$?