I am trying to solve the following exercise:
Let $k$ be an infinite field and let $f \in k[x_1,...,x_n]\setminus\{0\}$. Define $A = k[x_1,...,x_n, f^{-1}]$ as subring of $k(x_1,...,x_n)$. Find a Noether normalization of $A$.
I am not sure if my approach is correct, but I think I first need to check if $x_1,...,x_n, f^{-1}$ are algebraically dependent, which they are, because if I define $g \in k[y_1,...,y_{n+1}]$ as $g=f(y_1,...,y_n)y_{n+1}-1$ then $g(x_1,...,x_n, f^{-1})=0$.
Is this correct? And what do I do now?