On 9/19/07, John Cremona <[EMAIL PROTECTED]> wrote: > > I was intending to make the same suggestion myself (concerning Allan > Steele's method). When I read Allan's paper on this several years ago > I made a few suggestions to him on what I thought at the time might be > improvements -- which I will try to recover if this suggestion is > followed up.
Just out of curiosity do you know anybody who has ever actually made genuine use of Allan's QQbar implementation in Magma? I've played around with it myself, but never found a way to use it for anything interesting. I'm curious. Regarding actual implementation of algebraic number theory in Sage, it is going to be first primarily about replicating and extending the main critically needed functionality for Sage -- e.g., most things listed in the first few chapters of the extensions section of the Magma manual, before worrying about the sorts of things that are in being discussed in this thread. That said, it is extremely valuable that we are having this discussion, and I very much hope it will continue. Thanks, William > > Robert's suggestion that there is an obvious choice of root is all > very well for sqrt(2) but will not extend to more complicated > expressions, I fear. I don't understand or believe this, so could you elaborate? To explain why I'm confused, for arbitrary expressions involving only radicals (which is all Robert was talking about) I think it works fine, since there is a well-defined way to choose an n-th root for any n. Just choose the n-th root closest to the positive real axis with nonnegative imaginary part (I think that's what Maxima does). See it works: sage: a = (sqrt(2) + 5^(1/7))^(1/3) + (3+I)^(1/5) sage: a (I + 3)^(1/5) + (5^(1/7) + sqrt(2))^(1/3) sage: a.n() 2.64408949250617 + 0.0809560904915396*I sage: a.n(100) 2.6440894925061667374985582133 + 0.080956090491539622959385086896*I So long as you can always get a numerical approximation to arbitrary precision, you can construct a number field defined by them. Note that we're only talking about radical and not general roots of polynomials here. > > John > > On 9/18/07, John Voight <[EMAIL PROTECTED]> wrote: > > > > For inspiration, it might be worth comparing to Allan Steel's > > algebraically closed field construction: > > http://magma.maths.usyd.edu.au/magma/htmlhelp/text702.htm > > At no point is the field actually algebraically closed--it is just the > > affine algebra on the elements that you've already adjoined--but when > > you adjoin a new element it compares it to the existing one and > > inductively builds from there... > > > > John Voight > > Assistant Professor of Mathematics > > University of Vermont > > [EMAIL PROTECTED] > > [EMAIL PROTECTED] > > http://www.cems.uvm.edu/~voight/ > > > > > > > > > > > > -- > John Cremona > > > > -- William Stein Associate Professor of Mathematics University of Washington http://wstein.org --~--~---------~--~----~------------~-------~--~----~ 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/ -~----------~----~----~----~------~----~------~--~---
