Take the polynomial $f = x_1 x_2 + x_2 x_3 + x_3 x_1 \in k[x_1,x_2,x_3]$, and define the set $V = Z(f) = \{(x_1,x_2,x_3) \ | \ f(x_1,x_2,x_3) = 0 \} \subset \mathbb A ^3$. Consider the coordinate ring of $V$, given by $k[x_1,x_2,x_3]/(f) \cong k[a_1,a_2,a_3]$, where $a_i = x_i \ \mathrm{mod} \ (f)$.
Noether normalisation says there exists $ \{y_1, \ldots, y_m | \ m \leq 3\} \subset k[a_1,a_2,a_3]$ such that the $y_i$ are algebraically independent over $k$ and that $k[a_1,a_2,a_3]$ is a module-finite $k[y_1, \ldots y_m]$-algebra. If I can find such $y_i$, then I can explicitly construct a morphism $\phi : V \to \mathbb A^m$ that is surjective and has finite fibres (this is the geometric interpretation of Noether normalisation).
My question is: how do I go about finding such $y_i$?
Thanks