Noether's Normalization Lemma on $\mathbb{C}[x,y,z]/(xy+xz+yz)$: how to prove the algebraic independence?

Question

I'm trying to apply Noether's Normalization Lemma to $A = \mathbb{C}[x,y,z]/(xy+xz+yz)$. Following the Lemma's proof in Reid's "Undergraduate Commutative Algebra", I've reached that for $s = y-z^2$ and $t = x-z$, we've got that the extension $\mathbb{C}[s,t] \subseteq A$ is finite and that $z$ is integral over $\mathbb{C}[s,t]$ via the monic polynomial $$ p(\alpha) = \alpha^3 + \dfrac{1+t}{2}\alpha^2 + \dfrac{2s+t}{2}\alpha + \dfrac{ts}{2} \in \mathbb{C}[s,t][\alpha] $$ But to conclude, I have to prove that $s$ and $t$ are algebraically independent over $\mathbb{C}$. I think that it is possible by proving that the natural morphism $\phi:\mathbb{C}[X,Y] \to \mathbb{C}[s,t]$, with $X \mapsto s$ and $Y \mapsto t$ has $\ker \phi = \{0\}$, but I don't know how or even whether is the right way to do it.

Answer 1

I would map $A$ to $\Bbb C(X,Y)$ (the field of fractions of $\Bbb C[X,Y]$) via $x\to X$, $y\to Y$ and $z\to-xy/(x+y)$. The image of $A$ contains $X$ and $Y$ which are algebraically independent over $\Bbb C$, therefore so are $x$ and $y$ in $A$.