I am trying to compute an integral basis for the algebraic extension $K(z,y)$ of $K(z)$ by $y$, with $f(z,y)=0$, $$ f(z,y) = y^4-2zy^2+z^2-z^4-z^3 = 0. $$ $K$ here is either $\mathbb{Q}$ or $\mathbb{C}$, as convenient.
I am not really comfortable with the terminology, or sure about what I should do. I think I know that having an integral basis $\{w_k\}$ for $K(z,y)$ means that any function in $K(z,y)$ that has no poles can be written as a linear combination of $w_k$. So then I think that $$ \{1, y, y^2/z, y^3/z\} $$ is a basis, because it does not blow up anywhere, $w_k/z$ does, and it spans $K(z,y)$ when coefficients are in $K(z)$. I am not sure how to prove it actually is an integral basis for the integral closure of $K[z,y]$ in $K(z,y)$.
The problem is that I am not sure this is correct, and I know I need an integral basis before proceeding with the rest of the algorithm I am trying to carry out. (I.e., I know I've made a mistake somewhere, and it could be here.) The algorithm is the Risch-Hermite-Trager algorithm for integration of algebraic functions.
Any advice, please?
