Would someone be able to define a number field by supplying the value of the klein j-function at a point in a quadratic order? E.g. QQ[sqrt(-47), ellj((1+sqrt(-47))/2))]
What about: f1(x) = eta(x/2,1)/eta(x,1) K = QQ[abs(f1((1+sqrt(-47))/2))^2/sqrt(2)] What about values of the exponential function? QQ[exp(2*pi*i/37)] Another question is, should the field QQ[sqrt(2), sqrt(3)] be determined in a different way to QQ[sqrt(2) + sqrt(3)]? By the way, if it is decided that minimal equations of symbolic expressions should be determined using algdep, beware that one needs to compute the values of the actual algebraic expressions involved to as many digits of accuracy as one is requiring the equation to give. For example: \p100 algdep(2^(1/3)+sqrt(3),6,1000) will give the wrong answer. But then again, so will \p100 algdep(sqrt((2^(1/3)+3^(1/5))/2),30,100) Bill. On 18 Sep, 07:55, 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 --~--~---------~--~----~------------~-------~--~----~ 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/ -~----------~----~----~----~------~----~------~--~---
