On Tue, Jan 20, 2009 at 9:57 AM, Carl Witty <carl.wi...@gmail.com> wrote: > > On Jan 19, 7:50 pm, William Stein <wst...@gmail.com> wrote: >> Any commutative algebra computer computation over QQbar can be reduced >> to a computation over a specific absolute number field, since there >> are only finitely many symbols in the input to the calculation. >> Singular, etc., fully supports doing commutative algebra over absolute >> number fields. I think Singular doesn't explicitly support relative >> extensions, but might be able to do them anyways by just throwing in >> more variables. So if you're asking whether computer algebra systems >> such as Singular, Macaulay 2, Magma, etc., implement commutative >> algebra over QQbar the answer is formally "yes", in that there is a >> direct translation of any problem over QQbar into a problem they are >> designed to solve. > > I don't remember any features in Singular that would let you handle > relative extensions in any reasonable way; I think it only deals with > aboslute fields.
I was alluding to the fact that, e.g, one has an isomorphism of rings QQ(sqrt(2),sqrt(5))[x] ---> QQ(sqrt(2))[y][x] / (y^2 - 5) Thus any problem in a relative extension involving commutative algebra can be reduced to one with more variables and an ideal. William > >> Sage and probably the other systems are probably VERY awkward for >> actually doing computations over QQbar. There's a lot of bookkeeping >> and stuff that one would have to do, and conceptually it would be >> painful. Also, being able to do relative extensions is critical, >> since e.g., your coefficients could be the square roots of the primes >> up to 1000, and as an absolute field they would generate an extension >> of degre 2^168, which is too big. > > Currently, QQbar always computes absolute fields internally; being > able to get elements of QQbar in terms of relative extensions would > require a major rewrite of QQbar, unless there's some sort of trick > I'm not thinking of. Such a rewrite could be a good thing, but I > probably won't do it. I'd be happy to answer questions, etc., if > somebody else wanted to try, though. > > Carl > > > > -- William Stein Associate Professor of Mathematics University of Washington http://wstein.org --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to sage-support@googlegroups.com To unsubscribe from this group, send email to sage-support-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-support URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
