On Wed, Jun 24, 2009 at 9:15 AM, rje<[email protected]> wrote: > > > > On Jun 24, 4:35 am, David Joyner <[email protected]> wrote: > ... > >> At first I thought it should be 0 in this case because I >> confused "trivial character" with "identically equal to 1". > > That makes it surprising that T.jacobi_sum(T) was correctly defined to > be p-2 > (instead of p) for the trivial character T mod p.
I may be mistaken in my vague recollection. I didn't write the code but thought I remembered reading it over at one point. > > >> I don't understand why >> >> sage: parent(Z.jacobi_sum(Z)) >> Cyclotomic Field of order 20 and degree 8 >> >> is incorrect though. > > EVERY Jacobi sum over GF(5) lies in a much smaller cyclotomic field, > namely the field of Gaussian integers. That's because each term of the > defining sum is a Gaussian integer. For example: > > sage: parent(Z(3)) > Cyclotomic Field of order 4 and degree 2 This makes sense but (a) it seems that the parent might come from the coercion of some of the terms in the sum (which end up canceling) so IMHO the output is not wrong, (b) there might be a way to determine the smallest field in which Y.jacobi_sum(Z) lies but I don't know what it is, (c) one could argue that, in any case, that issue deserves a separate ticket (and patch). I just posted a patch to #6393, which needs review. > > rje > > > > --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to [email protected] To unsubscribe from this group, send email to [email protected] For more options, visit this group at http://groups.google.com/group/sage-support URLs: http://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
