On 20 Sep, 08:22, Robert Bradshaw <[EMAIL PROTECTED]>
wrote:
> > Yeah perhaps radicals should just be meaningless unless you have a
> > number field embedded in C, in which case they should just return the
> > root with the least argument.
>
> I think it should give something, but the choice might be arbitrary,
> and one would be able to make the choice manually if one wanted.
Well the important thing is that in the abstract number field setting,
a radical should only return a result if the element you are taking
the nth root of is an nth power in the number field. But with no
embedding into the complex numbers to refer to, how do you make a
choice of nth root? You could just return a random nth root I suppose.
Either that, or you have to return all the possible values. So long as
the user can do both, I suppose there is no harm.
I personally don't like the idea of my CAS making arbitrary choices
for me where there is no convention on which to make those choices.
But that is just me. So long as both options exist, I guess it will
all be good.
I believe I recall an algorithm based on Hensel lifting which
efficiently finds roots in number fields, should they exist.
I'm thinking that adjoin(K, y^3-2*y+7) is not so stupid after all. But
I still didn't think of an algorithm. I think Pari implements one
though via its function rnfequation. You can define a relative number
field by a polynomial pol over a number field nf and then ask it to
give an absolute equation apol. It even supplies an equation for the
generator of nf in terms of a root of apol. Nice.
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/
-~----------~----~----~----~------~----~------~--~---
