2009/9/3 javier <vengor...@gmail.com>
>
> Hi William,
>
> On Sep 3, 8:18 am, William Stein <wst...@gmail.com> wrote:
> > I am not sure what something like "integers with I adjoined" is?
>
>
> I guess that means the complex numbers of the fomr a + bI with a, b
> integers, or Z[I] (the Gaussian Integers). Mathematica prides itself
> to be able to apply primality tests, factorization algorithms and the
> kind in this ring:
> http://reference.wolfram.com/mathematica/ref/GaussianIntegers.html
>
> Maybe we should add an easy way of working with it, taking advantage
> of Sage fast arithmetic for integers?
>
Sage has the Gaussian integers, and I'm sure the basic arithmetic and
functionality is as good or better than Mathematica already.
sage: R.<I> = ZZ[sqrt(-1)]; R
Order in Number Field in I with defining polynomial x^2 + 1
sage: a = 2 + 3*I
sage: timeit('a*a')
625 loops, best of 3: 1.4 µs per loop
sage: R.ideal(3)
Fractional ideal (3)
sage: R.ideal(3).factor()
Fractional ideal (3)
sage: R.ideal(5).factor()
(Fractional ideal (-I - 2)) * (Fractional ideal (2*I + 1))
sage: R.class_group()
Class group of order 1 with structure of Number Field in I with defining
polynomial x^2 + 1
sage: P = R.ideal(11); P
Fractional ideal (11)
sage: P.is_prime()
True
sage: k = R.quotient(P,'a'); k
Quotient of Maximal Order in Number Field in I with defining polynomial x^2
+ 1 by the ideal (11)
Of course Sage can do all the above sort of stuff with any number field.
William
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