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All four of those factors are prime in the new number ring, giving 6 a dual existence Those factors are irreducibles, but they aren't primes, which is the rea
by ginnungagap 9y ago
All four of those factors are prime in the new number ring, giving 6 a dual existence
Those factors are irreducibles, but they aren't primes, which is the reason why the uniqueness of factorization fails.
A nonzero non unit element of a ring is called prime if p|ab implies p|a or p|b.
A nonzero non unit element of a ring is called irreducible if p=ab implies that a is a unit or b is a unit (invertible element).
Primes are irreducibles in an integral domain, but the converse is true in unique factorization domains and Z[√-5] is not one.
- wodenokoto 9y agoThat'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 ...
- umanwizard 9y agoThis isn't journal article level stuff; if you are interested, pick up any undergraduate book on abstract algebra and start working through it.
- moomin 9y agoIt’s also kind of irrelevant: what they’re called isn’t really the point.
- coldtea 9y agoActually it is relevant, since math is syntax and semantics. If we get the semantics wrong it's gibberish.
- ginnungagap 9y agoWhat 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
- ginnungagap 9y agoWikipedia 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 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.
- gggvvh 9y ago> Wikipedia isn't a good source Why would you say that?
- SomeStupidPoint 9y agoIt'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).