Let $f=-y^n+x^{n-1}-1$ and $U=Z(f)$. Show that for all $P \in U \cap D(x)$ where $D(x)= \{(x,y) \in \mathbb{A}^2| x \neq 0\}$, $y-y(P)$ is a parameter or uniformiser at $P$.
We have $\dfrac{\partial f}{ \partial x}(P) \neq 0$ for all $P\in U \cap D(x)$. To indicate $y-y(P)$ is a parameter, i want to show that $y-y(P) \notin \mathfrak{m}_P^2$ where $\mathfrak{m}_P$ is an maximal ideal of $P$ in $\mathcal{O}_X(U \cap D(x))$. Now we have $$\mathcal{O}_X(U \cap D(x))= \dfrac{k[x,y,z]}{(zx-1,-y^n+x^{n-1}-1)}$$ How can we find the maximal ideal in this quotient ring?