Let $K/F$ be a field extension, let $α ∈ K$ be an algebraic element with minimal polynomial $f(X) ∈ F[X]$, and let $r ∈ F^\times$. What is the minimal polynomial for $rα$ in terms of $r$ and $f$?
I think I am getting mixed up but I cannot see a way to do this problem without expressing the minimum polynomial for $r \alpha$ in terms of $r,f,\alpha$. I have tried rearranging it many times and has come of no avail.
Ok, now having thought a little more, we can see that the minimal polynomial must be $g(\cdot)=f(\frac{\cdot}r)$ as clearly $g$ has $r\alpha$ as a root and if it weren't the minimal polynomial then $f$ would not be the minimal polynomial of $\alpha$ (essentially by the same trick done the other way on any lower degree minimal polynomial of $r\alpha$).