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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?

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    $\begingroup$ Second-last line: "'the prime p is in the prime decomposition of (q-p)⁢c and thus also at least of q-p or c.  But we know that  p∤c, whence  p∣q-p." Are you using Euclid Lemma here, or am I missing something? $\endgroup$ – Francesco Polizzi Sep 4 '19 at 14:53
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    $\begingroup$ Since $n_0 < n$, by induction hypothesis $p$ must appear in the unique decomposition of $n_0 = (q-p)c$, which is the product of the unique decompositions of $q-p$ and $c$. It's true that the sentence is quite confusing. $\endgroup$ – Wille Liou Sep 4 '19 at 17:36
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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.

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Note that the popular textbook by Niven, Zuckerman, and Montgomery also has a proof of unique factorization in $\mathbb{Z}$ that is based only on the well-ordering property and that bypasses Euclid's lemma.

The ascending chain condition in the definition of Dedekind domains is used in a similar way to the well-ordering principle in proving unique factorization of ideals in such domains.

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