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?
John
On 14/12/2007, William Stein <[EMAIL PROTECTED]> wrote:
>
> On Dec 13, 2007 3:20 PM, Robert Miller <[EMAIL PROTECTED]> wrote:
> > The algebraic definition is that its only divisors are itself and 1,
> > up to units-- yes. The cultural definition is that it is in {2,
> > 3, ...}-- no. It seems weird to use the cultural definition, since if
> > we pass to another ring that contains ZZ, chances are the definition
> > there will be the cultural definition. If -7 goes from not prime to
> > prime when we extend, say, to the ring of integers of a number field,
> > then that feels weird. Anyway, the ticket is question is:
> >
> > http://trac.sagemath.org/sage_trac/ticket/1399
>
> Currently there is no is_prime method on elements of rings of integers or
> multivariate polynomials. There should be an is_irreducible
> method in general (not is_prime). I think is_irreducible should
> allow for negative elements, and be the general notion. I don't
> like is_prime being used for that general notion.
>
> Here's the typical sort of careful quote you find all over in wikipedia
> and other references: "It is helpful to compare irreducible
> polynomials to prime numbers: prime numbers (together with the
> corresponding negative numbers of equal modulus) are the irreducible
> integers." It's quite clear that "prime" refers specifically
> to positive integers.
>
> -- William
>
> >
>
--
John Cremona
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