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What are direct methods for proving that a ring is a UFD in general without proving that it's a PID/Euclidean domain/field and using the fact that all those things are UFDs?

As an example, we can take $\mathbb{Z}[i]$ or $\mathbb{Z}[\sqrt{-2}]$ or other rings you come up with.

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    $\begingroup$ Seems to me that if you're interested in this for some curious reason you might ask first how to show that $\Bbb Z$ is a UFD by another method... (no, you don't necessarily see the words "suclidean domain" in the standard proof, but that is how the standard proof for $\Bbb Z$ goes.) $\endgroup$ – David C. Ullrich Apr 9 '18 at 14:59
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    $\begingroup$ You might want to look at a PID (hence a UFD) which is not an euclidean domain, like here. Think also about $\Bbb Z[X]$ : it is not euclidean (not even a PID), but it is a UFD (however euclidean divisions by monic polynomials always work). $\endgroup$ – Watson Apr 10 '18 at 7:14
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Here are some thoughts:

  • Exsitence is easy to prove using induction on the norm.

  • Uniqueness is the hard part, especially since it fails for most rings of the form $\mathbb Z[\sqrt d]$. For the rings you've mentioned, it can be proved by knowing the units and exactly how primes in $\mathbb Z$ decompose in $\mathbb Z[\sqrt d]$. There are three possibilities for a prime $p$: it remains prime, it is product of two non associate primes, it is a square. For the rings you've mentioned, this can be decided in ad hoc ways.

In the general case of the ring of integers in quadratic fields $\mathbb{Q}(\sqrt{n})$, the answer is not simple but is fascinating, See the book Primes of the Form $x^2+ny^2$, by David Cox.

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This would be extremely wasteful, and nobody would do it before showing they were Euclidean domains or PIDs, but you could show that they have class number 1 through some indirect means. $\mathbb Z[i]$ is a Dedekind domain so it is a UFD if and only if its class number is 1.

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  • $\begingroup$ What about UFD s that aren't even PID s? $\endgroup$ – John Cataldo Apr 9 '18 at 15:17
  • $\begingroup$ That's a god question, and I'm not sure there's a general method. I mean, there are things like that if $R$ is a UFD, then so is $R[x]$, but that's not useful in many circumstances. $\endgroup$ – Jordan Hardy Apr 9 '18 at 17:33

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