Hi Christian,
On Tue, Jun 08, 2010 at 03:20:27PM -0700, Christian Stump wrote:
> Another question I just thought about was: Do we want:
>
> 1. every element in the universal cyclotomic field living in exactly
> one cyclotomic field QQ( \zeta_n ) generated by ZumbroichBasis(n,1)
> for some n, or
> 2. can an element have several monomials living in different
> cyclotomic fields.
>
> In gap, it is the first (as there is no universal cyclotomic field but
> only smart back and forth going within several CFs: e.g.,
> gap> E(8)^2;
> E(4)
> gap> E(8)+E(8)^2:
> E(8)+E(8)^2
> whereas in a not redundant basis expression in the universal
> cyclotomic field, the second would answer E(8) + E(4). I personally
> would prefer the first, as I think is quite annoying if one has longer
> expressions with different E(n)'s involved. What do you think?
I don't have first hand experience with the universal cyclotomic
field, so I don't know. Jean-Michel and Andrew Mathas may have a point
of view on this; I'll ask them next week. In the mean time, and at
first sight:
1. seems indeed easier to read in many situations, although not
always. Imagine an expression involving I = E(4), and an n-th root
of unity with n large. Then it will be hard to tell them apart.
2. seems easier to implement, and as a consequence more
efficient. The point is that one can really work locally
(i.e. term by term), and yet globally have a normal form (e.g. no
conceptual linear dependency between basis elements).
For 1, do you see other advantages beside user readability? That is,
for an algorithmic point of view when *using* such elements, would
1. make things easier? Otherwise, one could go for 2, and just have
_repr_/_latex_ make sure that the *output* would only involve a single
E(n).
Cheers,
Nicolas
--
Nicolas M. Thiéry "Isil" <nthi...@users.sf.net>
http://Nicolas.Thiery.name/
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