Let $K \subset L$ be fields and $\tau \in L$ be transcendental over K. Then $\tau$ is algebraic over $K(\alpha)$ for any $\alpha \in K(\tau) - K$ Let $K \subset L$ be fields and $\tau \in L$ be transcendental over K. Show that $\tau$ is algebraic over $K(\alpha)$ for any $\alpha \in K(\tau) - K$
Any help with this question? I'd love a hint instead of a solution!

An element in $K(\tau)-K$ is a rational function in $\tau$, so there are polynomials $f,g$ with coefficients in $K$ such that
$$\alpha=\frac{f(\tau)}{g(\tau)}$$
Can you make a polynomial with a root of $\tau$ from this?
