Salut Nicolas!
> 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).
The good thing about the Zumbroich basis is that if you have an
element in the cyclotomic field CF( n ), it is easy to decide in which
smallest subfield CF( m ) with m | n the element actually lives. The
Zumbroich basis does not provide a basis for a universal cyclotomic
field (as far as I know, but I don't have Zumbroich's thesis but only
another article citing and describing it!). I think I could use it to
construct a basis for the UCF but I doubt that this would be handy to
use (as the dependencies I would have to eliminate are a little
tricky). The basic idea is the following: we know that \sum_{ 0 \leq k
< n } E(n)^k = 0 <==> \sum_{ 1 \leq k < n } E(n)^k = -1. Multiplying
both sides with E( n' ) for any n', we obtain dependencies between
Zumbroich bases for different n's, which I could solve only
artificially.
So if you want to multiply two elements where the first element lives
one cyclotomic field CF( n ) and the other lives in CF( n' ), then it
is easy to embed both into CF( lcm( n, n' ) ), multiply them and then
(easy in the Zumbroich basis) push it down to the smallest subfield
CF( m ) with m | lcm( n, n' ) i which it lives in.
If you want to multiply two elements both having monomials in
different CF's, then you have to push them anyway into the same
CF( lcm( n_1, n_2, ..., n_k ) ) with n_i being the different CF's
involved, multiply them there and then push them down again. If you
want to write the product in a "universal Zumbroich basis", the
pushing down becomes, as described above, much more involved.
Maybe you can discuss it with some people next week and then let me
know what they think...
Best_1, Christian
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