On Mar 16, 2007, at 1:36 PM, Joel B. Mohler wrote:
> I'm trying to think about the math a little bit behind this too. > It seems that > I shouldn't need a denominator. Are there number fields that need > to be defined > in terms of a polynomial with rational coefficients? Or, does it > have to do > with the fact that the number field code insists on the polynomial > being monic? All number fields can be defined in terms of a monic polynomial with integer coefficients. Suppose you start with a defining polynomial which is monic but has non-integral coefficients. Multiply the whole polynomial by L^d, where d is the degree and L is the LCM of the denominators of the coefficients. Then you can absorb L into the variable. (Another way to say this is to take any generator of K/k and then multiply it by an appropriate integer to make it an algebraic integer.) On the other hand, the user should be allowed to make number fields by polys with non-integral coefficients. Perhaps you want to translate for them, to make the arithmetic fast and the user experience transparent. Sounds like a pain though. Of course to represent *elements* of the number field, you need denominators. Probably best to represent elements using a polynomial in Z[x], with a single denominator, which is I think what you've been suggesting. David --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to [email protected] 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/ -~----------~----~----~----~------~----~------~--~---
