Quadratic cone invariant ring
= Quadratic cone invariant ring
{title2=$\mathbb C[a,b,c]/(ac-b^2)$}
The scalar sign action of a <cyclic group> of order two on $\mathbb C^2$ has invariant ring $\mathbb C[X^2,XY,Y^2]\cong\mathbb C[a,b,c]/(ac-b^2)$. It is not factorial: $a,b,c$ are pairwise nonassociate irreducibles, since every nonconstant invariant has degree at least two, while $ac=b^2$ gives two distinct factorizations.