4 ms·
I think the TLDR is: We could but it's inconvenient, so we don't usually allow that. Specifically, allowing negative numbers (or the number 1) to be primes sc
by FartyMcFarter 6y ago
I think the TLDR is:
We could but it's inconvenient, so we don't usually allow that.
Specifically, allowing negative numbers (or the number 1) to be primes screws up the fact that every natural number otherwise has a unique factorization.
- deleted 6y ago[deleted]
- ogogmad 6y agoAllowing 1 to be prime is a lot worse. The integers modulo 1 don't form a field, but modulo a negative prime they do. I think algebraists do generally consider negative primes to be primes.
- pmiller2 6y agoIt's even worse than that. Because integers a and b are congruent modulo n iff n | a - b, and 1 divides every integer, the integers modulo 1 has only 1 element. This gives you the somewhat pathological object called the zero ring: https://en.wikipedia.org/wiki/Zero_ring https://en.wikipedia.org/wiki/Zero_ring
- pmiller2 6y agoNot really. Unique factorization is defined up to units, so, factors of +-1 don't matter. 1 is not a prime because it's a unit, and the ideal it generates is trivial; it's the entire ring, so, for Z, it's all of Z.
- FartyMcFarter 6y agoThe fundamental theorem of arithmetic doesn't say anything about units though: https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithme... > every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors.
- ogogmad 6y agoLook up the definition of a Unique Factorisation Domain and a prime element of a ring. The theorem that the integers form a UFD necessarily takes into account negative primes. [Edit] Here: https://en.wikipedia.org/wiki/Unique_factorization_domain https://en.wikipedia.org/wiki/Unique_factorization_domain