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Comment by wodenokoto | Hacker News Reader

That's a statement that goes quite against the claims of the article and I kinda want to ask for a citation ... But I already feel that this discussion is way over my head and I wouldn't understand any journal articles that would clear this up ...

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Wikipedia isn't a good source, but it uses 3 as an element of Z[√-5] as an example so it's quite relevant: https://en.m.wikipedia.org/wiki/Irreducible_element

For a more serious reference any abstract algebra book covering rings and UFDs should do, for example it is in Dummit & Foote.

This isn't journal article level stuff; if you are interested, pick up any undergraduate book on abstract algebra and start working through it.

It’s also kind of irrelevant: what they’re called isn’t really the point.

Actually it is relevant, since math is syntax and semantics. If we get the semantics wrong it's gibberish.

What they are called might not be the point, but calling them prime in a context in which prime has a precise meaning while they are not can be confusing. Especially since the difference between prime and irreducible is subtle and it's the whole reason we have non unique factorization domains, which is the main topic of the article

It's mostly just pointing out that in a particular extension of the integers, if we attach the square root of -5, then (1 + sqrt-5) * (1 - sqrt-5) = 2 * 3.

3 divides the left hand side, but it is not a divisor of either term. This is different than in the usual integers, since there are irreducible terms (not divisble by anything) that aren't prime (if they divide a product, they divide one of the factors).

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