We know the ring $ \mathbb{Z}[\sqrt{-5}] $ is not a UFD.
The typical proof is showing that $6$ factors in $2$ ways.
But there must be a better way to show this than to do trial and error factoring of some random element of the ring.
So how does one prove $ \mathbb{Z}[\sqrt{-5}] $ is not a UFD without factoring a certain element ?
I can show it must be atomic and has a finite amount of units. But that is insufficient and perhaps even irrelevant ; UFD have that too.