Exercise from "Abstract Algebra: An Introduction" by T.W.Hungerford.
Let $K$ be an extention field of $F$. If $u,v \in K$ and $u+v$ is algebraic over $F$, prove that $u$ is algebraic over $F(v)$.
This should not be an exhaustive exercise, however, I am stuck with it. I know that "if" part in the statement of the exercise simply tells that $f(u+v)=0$ for some $f(x)\in F[x]$, but how can I prove that also $g(u)=0$ for some $g(x)\in F(v)[x]$? Hints appreciated.