We need to Find a non constant map $f:\mathbb{C}^3\to \mathbb{C}$ such that for any three distinct complex numbers $z\_1,z\_2,z\_3$ and any automorphism $\phi$ of $\mathbb{C}$, we have
$f(z\_1,z\_2,z\_3)= f(\phi(z\_1),\phi(z\_2),\phi(z\_3))$


Thank you for help and discussion.



to find a  function with a property
If the $z\_i$s are all distinct, take


$f(z\_1,z\_2,z\_3) = \frac{z\_1-z\_2}{z\_3-z\_2}$


This is the cross-ratio. It can take any complex value except for $0$ and $1$. It is clearly invariant. You can get many other functions by composing this with some function from $\mathbb C - \{0,1\}$ to $\mathbb C$. Indeed, these functions are all the possible functions, since if two triples have the same cross-ratio, they are the same up to an automorphism.


