It's ok for sqrt(2).parent() to be an order, but what about sqrt(1/2).parent() ? Would that be the field? Not very nice since it will not be obvious from the form of the input whether or not the symbolic expression is integral.
There is a whole can of worms here just waiting to be released, under the heading "nested radicals". Having said that, I do like the idea of generating number fields (as opposed to orders) by giving generators in a natural way. John On 9/18/07, Robert Bradshaw <[EMAIL PROTECTED]> wrote: > > On Sep 17, 2007, at 9:00 PM, William Stein wrote: > > > This is being cc'd to sage-devel, since no reason not to. It's me > > and Robert Bradshaw working on reworking the algebraic number > > theory code in Sage (we've done a lot now). > > > > On 9/17/07, Robert Bradshaw <[EMAIL PROTECTED]> wrote: > > BTW, I've been working on quadratic number field elements... > > > > That's a good idea. > > I have been working on grant proposals all day long. > > I'm going to switch gears and work on the ANT package > > soon. I'll probably work only on getting all the doctests > > not in the number_field directory to pass, since much > > was broken by my changes. > > > > I also want to make ZZ[a,b,c] > > work, if a,b,c are algebraic integers. > > > > It would also be really neat to have a function that can > > compute the minimal polynomial of a symbolic element: > > sage: a = sqrt(2) > > sage: a.minpoly() > > x^2 - 2 > > sage: a = 5^(1/3) > > sage: a.minpoly() > > x^3 - 5 > > > > One possibility would be to numerically approximate a, > > use pari's algdep to get a candidate minpoly f, then > > do bool(f(a) == 0). If it works, we're golden. If not, > > we give up. Since bool(f(a) == 0) errors on the side of > > caution, this would probably be fine. > > That sounds like a really slick idea. > > > With that, we could do > > > > ZZ[sqrt(2), 5^(1/7), sqrt(7)] > > > > and it would work. Thoughts? > > I think this would make for a really natural way of constructing > number fields. I am still of the mind that I would like sqrt(2).parent > () to be an order in a number field (with an embedding into C > choosing the positive root), assuming the coercion model was robust > enough to find resonable compositums of these things. > > - Robert > > > > > -- John Cremona --~--~---------~--~----~------------~-------~--~----~ 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/ -~----------~----~----~----~------~----~------~--~---
