Hi, I want to ask you some questions concerning the arithmetic of number fields in Sage.
Does Sage currently support factoring in Gaussian integers (i.e. in the ring Z[I] of complex numbers with integral real/imaginary parts)? I thing this would be a nice feature to have. In pari/gp for example this works: ? factor(5*I) %1 = [2 + I 1] [1 + 2*I 1] in sage factor(5*I) gives an error. Obviously factoring depends on the ring in which we are working on, so that for example the factorization of 5 would be different in the ring Z or in the ring Z[I] Currently Z[I] seems not to be a supported ring in sage, even though you can create a quadratic field F=QuadraticFiled(-1,'a') (number field in a with defining polynomial x^2+1) and after doing a= F.gen() 'a' behaves algebraically as the complex number I (i.e. a^2=-1) [ Mathematically 'a' is not the complex number 'I', as it could be also the complex number '-I' wich is also a root of the same minimal polynomial x^2+1 ; however there exists the function complex_imbedding() that gives the corresponding complex number: sage: a.complex_embedding() 1.00000000000000*I assuming that a=I) but there seems to be no implementation of factorization in quadratic fields (of course in those fields in wich there is a unique decomposition into primes), nor a notion of the subring of algebraic integers. My question is: what would be the best way to implement this? Perphaps the best way would be creating a new ring sage.ring.complex_integer representing gaussian_integers for representing Gaussian integers I see that currently a complex number like 2+3*I is represented in sage, by using float point representation. z=2+3*I sage: type(z) <type 'sage.rings.complex_number.ComplexNumber'> sage: z.real() 2.00000000000000 sage: type(z.real()) <type 'sage.rings.real_mpfr.RealNumber'> sage: and this could lead to rounding errors. It would be better to do exact integer arithmetic, when dealing with Gaussian integers. best regards Pablo --~--~---------~--~----~------------~-------~--~----~ 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/ -~----------~----~----~----~------~----~------~--~---
