Algebraic dependence of $xy, xz$ and $ yz$ over $\mathbb{C}$ and their symmetry

Question

1. Show that $ xy, xz, yz \in \mathbb{C}[x,y,z]$ are algebraically dependent over $\mathbb{C}$ by finding a polynomial they satisfy. Is there a more general method or approach that I should take when looking for this kind of algebraic dependence?
2. Does the symmetry between $xy, xz $ and $ yz $ imply that the polynomial they satisfy must be symmetric? How to prove it without explicitly finding the polynomial?

Answer 1

They are not algebraically dependent over $\mathbb{C}$! Probably the easiest way to see this is to think about the subfield $\mathbb{C}(xy,xz,yz)\subseteq \mathbb{C}(x,y,z)$ which they generate, since using division we can isolate one variable at a time. Notice that $(xy)(xz)/(yz)=x^2$ is in this subfield, and so by symmetry so are $y^2$ and $z^2$. But $x^2,y^2,$ and $z^2$ are algebraically independent over $\mathbb{C}$, so the $\mathbb{C}(xy,xz,yz)$ has transcendence degree at least $3$ over $\mathbb{C}$. It follows that the three generators $xy,yz,xz$ must be algebraically independent.