Let $u \in \mathbb{K}(X), \ u = \frac{X^3}{X+1}$. Prove that $\mathbb{K}(X)\supset \mathbb{K}(u)$ is an algebraic extension and find $[\mathbb{K}(X):\mathbb{K}(u)]$.
My attemps were trying to show that is a finite extension. If I can prove that then automatically is an algebraic extension. What I have tried to do is to write every element of $K(X)$ as lineal combinations of $\{1,u,u^2,u^3,..,u^n\}$ but it seems impossible.
The minimal polynomial of $X$ over $\mathbb K(u)$ is seen by inspection to be $ z^3-uz-u, $ whereupon $\mathbb K(X)$ is a degree $3$ algebraic extension of $\mathbb K(u)$.
If this is confusing to you due to the fields containing indeterminates, think by analogy with a field extension of $\mathbb Q$ by a root of $z^3-z-1$.