On Dec 14, 2007 3:13 AM, John Cremona <[EMAIL PROTECTED]> wrote: > > I was going to reply when I saw that William had said almost the same > as I was going to. > > I am happy with is_irreducible(-7) being true but is_prime(-7) being > false. When we teach the distinction between irreducibles and primes > in more general integral domains we define prime elements in terms of > their divisbility properties (i.e. p|ab => p|a or p|b), show that > primes are always irreducible while irreducibles need not be prime. > But the upshot of that is that there are rings (e.g.Z) for which the > irreducibles and primes (as just defined) are the same thing, and > others where they are not -- and in these other rings the notion > "prime" is just not useful as an attribute for elements, so one goes > over to using it for ideals only. > > This means that the only ring in which there is still any need to talk > about prime elements at all, in practice, is Z; where we can follow > convention to only allow positive irreducibles (!) to be "prime". > > Conclusion: is_prime should be defined for integers, and ideals, but > need not be defined (i.e. implemented) for elements of any other ring > than Z. > > Does anyone agree?
I do. :-) Mike Hansen: > I think the best course of action would be the following. Integers > get a is_prime_number method with is_prime being an alias to > is_prime_number. I think that distinguishing between is_prime and is_prime_element is very very confusing. And I agree with John Cremona that "prime" is not a very useful notation in algebraic number theory / commutative algebra for *elements* -- it's a great notion for ideals. -- William --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-devel@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://sage.scipy.org/sage/ and http://modular.math.washington.edu/sage/ -~----------~----~----~----~------~----~------~--~---
