4 ms·
Unique factorization is always meant up to unities. You haven't really lost anything.
by giomasce 6y ago
Unique factorization is always meant up to unities. You haven't really lost anything.
- ot 6y agoHey Gio, long time no see :) The definition "up to unities" is generally when you abstract things and talk about unique factorization rings. Then your "primes" (formally irreducible elements) become equivalence classes up to unities. In the concrete integers though AFAIK it's more conventional to only consider positive primes, so that the decomposition is canonical. Probably I should have said that you lose canonicity.
- giomasce 6y agoHey, didn't notice the nick! Hi! Well, if you're speaking about non-negative integers, you don't have the problem of the primality of negative numbers. If you're considering relative integers, still you don't really have a unique factorization property, because negative integers do not have a factorization altogether (in terms of positive primes). So you have to stipulate that a prime factorization is not just a "product of primes", but a "product of primes, perhaps times -1", which is still not the way you usually define it. So I am not totally convinced. In the end, everybody just pick the definition they like.
- ot 6y agoI've definitely seen the definition x = u p_1 p_2 ... where u is any unity, but I agree that people pick the definition they need depending on the context.