There's so much interesting reading in this thread that I am having trouble reading it all between the talks of the conference I'm at...
This morning I heard a talk by Gunther Malle describing experiments on class groups of cyclic cubic fields (i.e. Galois extensions of Q of degree 3), where he had computed a very large number of these (over 10^7 certainly, I forget). He said he used pari/gp, and that therefore the results were conditional on GRH, but that it would have been impossible to have gone so far using anything unconditional (or to certify the pari/gp results within pari/gp) -- the discriminants went up to 10^22 I think. The point of mentioning this is that however much we might like Sage to only ever give certified results (for number field computations) it is also importatn to have the capability of doing conditional computations. Malle's work could not have been done by Sage if Sage's algorithm was pari+certification. John On 9/20/07, Robert Bradshaw <[EMAIL PROTECTED]> wrote: > > On Sep 20, 2007, at 12:05 AM, Bill Hart wrote: > > > On 20 Sep, 06:39, Robert Bradshaw <[EMAIL PROTECTED]> > > wrote: > > > >>> To define an abstract number field, something like K = > >>> number_field(QQ, x^3-3*x^2+1) could work. Eventually one will > >>> want to > >>> be able to do adjoin(K, y^5+7*y-1), compositum(L, M), etc. > >> > >> Would/could these functions return lists too, if, say, the > >> intersection of L and M was not Q? > > > > Yeah, thinking about it some more, I think these functions really only > > make sense if everything is embedded in another field. In particular > > they should make sense if everything is embedded in an algebraic > > closure of Q. > > Yes, but the idea is that the field both are embedded in could (in > some cases) be automatically generated, or there would be a finite > list of possibilities of compatibly embedding of L and M in to a > larger field. > > > But one might need to specify the field they are embedded in, since > > ideally number fields should be allowed to be specified with more than > > one embedding. > > > > I suppose that one could limit the number of embeddings of K to 1 and > > then just clone K and give the clone a different embedding. But > > whatever means is used to clone K it should be able to specify an > > isomorphism between the two copies so that one can easily go from the > > elements of one embedding field to the other. For example it would be > > nice to embed K into its Galois closure and then think of this as > > being identified with a subfield of Q bar. This would require K to be > > simultaneously embedded in Q bar and in the Galois closure of K. > > This is what I envision, and is more consistent with the philosophy > of SAGE that objects should be immutable. Given an abstract number > field K, one would then construct a number field K1 = K.with_embedding > (map_from_k_to_L) which would be a new object. I envision embedding > would be transitive, and arithmetic between K1 and L would result in > elements of L, whereas arithmetic between K1 and K would yield > elements of K. > > > Of course one's wish list always grows longer than one's lifespan and > > ability to implement. > > Can't argue with that! > > > > > -- John Cremona --~--~---------~--~----~------------~-------~--~----~ 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/ -~----------~----~----~----~------~----~------~--~---
