Field Extensions: Algebraic and Transcendental Elements

Hello there. I have the following problem:

Suppose that $\displaystyle v$ is algebraic over $\displaystyle K(u)$ for some $\displaystyle u$ in an extension field of the field $\displaystyle K$. If $\displaystyle v$ is transcendental over $\displaystyle K$, show that $\displaystyle u$ is algebraic over $\displaystyle K(v)$.

I've tried writing down the minimal polynomial for $\displaystyle v$ over $\displaystyle K(u)$ and playing around with that, but I haven't had any luck. I am also somewhat baffled by the transcendental assumption - where does it even come in to play, and how can I write it down? (I usually see it defined as a negative.)

Just a hint in the correct direction would be appreciated.