On Sep 18, 2007, at 7:42 AM, Bill Hart wrote: > 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)
If the inputs are symbolic, one can compute them to very high precision very quickly, and the really nice part is that one can then provably verify (by symbolic substitution) that one has indeed found the correct field. The biggest challenge seems to be guessing what the degree of the extension is (but one could conceivably try many composite degrees hoping for success on one of them). As to the question of embedding, I would (perhaps radically) propose that QQ[sqrt(2)] has the embedding sqrt(2)>0 (and in general QQ[a] has the embedding a -> CC(a) for any symbolic a), and if one wants the abstract number field, one would have to use the NumberField (x^2-2) construction. - Robert > > 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/ -~----------~----~----~----~------~----~------~--~---
