TITLE: Zermelo's proof for unique factorisation
QUESTION [7 upvotes]: In Peter Bundschuh's "Einführung in die Zahlentheorie" I came across a possibly well-known but to me rather peculiar proof of unique factorisation, which is attributed to Ernst Zermelo. The proof bypasses Euclid's lemma to prove that $ \mathbb{Z}$ is a UFD. It seems to me that the deepest property of the ring of integers used is some properties of the order on $ \mathbb{Z}$! An account of this proof can be seen here: 
https://planetmath.org/inductionproofoffundamentaltheoremofarithmetic
Can this idea be applied or has it been applied to any other situations? Or is it simply a tricky proof for the ring of integers?

REPLY [6 votes]: In Pete Clark's notes on factorization, he calls this the "Lindemann–Zermelo proof" of the fundamental theorem of arithmetic, and he uses a similar idea to prove the well-known fact that if $R$ is a unique factorization domain, then so is $R[t]$.  See Theorem 27.