The two things that Allan's method seems to allow that the embedding into C won't directly allow are:
1) Return an absolute number field which contains all the elements defined so far. 2) Compute a splitting field of a given field. Numbers 1 and 2 could be implemented independently though. Allan says number 1 is very expensive and probably not practical if the absolute degree of the resulting field is > 20. So no surprises there. Number 2 is also going to be expensive, but it is with Allan's system too. So for Qbar at least there is no advantage that I can see of doing things Allan's way. Bill. On 20 Sep, 17:38, Bill Hart <[EMAIL PROTECTED]> wrote: > No wait. I'm missing something important here. As you say, computing > Galois groups is a very hard problem (probably impractical for > polynomials of degree > 50 according to the Magma documentation). But > computing a Galois closure of a field is going to be equivalent, since > then one could compute the Galois group of the original polynomial by > looking at the action of the Galois group of the Galois closure on the > generator of the Galois closure. > > What you suggest is in fact going to be faster. Each element of the > algebraic closure will be specified as a minimum polynomial and a root > of that polynomial computed to sufficient precision to distinguish it > from the other roots of that polynomial. To compare two elements, one > first compares their minimum polynomials. If they are the same, one > then compares the roots. > > It is unfortunate that this forces the implementation to pick a > particular root at random when using radicals. > > The only question I now have is, what is the difference between this > idea and embedding a number field in CC? Is it just that roots are > automatically computed to sufficient precision to distinguish them > from their conjugates? > > I guess we need to look at the functions that Allan Steel's > implementation provides for Qbar and check that each of them can be > implemented with such a scheme. > > Bill. --~--~---------~--~----~------------~-------~--~----~ 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/ -~----------~----~----~----~------~----~------~--~---
