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/ -~----------~----~----~----~------~----~------~--~---
