On Nov 27, 6:12 am, Christian Stump <christian.st...@gmail.com> wrote: > sage: UCF.<E> = UniversalCyclotomicField(); UCF > Universal Cyclotomic Field endowed with the Zumbroich basis > sage: E(5) > E(5) > sage: UCF.<zeta> = UniversalCyclotomicField() > sage: zeta(5) > zeta5
OK, now we're firmly arriving in bikeshedding territory: Is there a reason they print differently? Does "zeta[5]" (both as printname and input notation) make sense? > UCF.gen(5) > zeta5 > UCF.gen(5,2) > zeta5^2 The difference between UCF.gen(0) and UCF.gen(1) is cute, but how much field code out there expects that K.gen(0) is a field element? If being cute at this point causes errors, incompatibility and limited use of UCF in other situations, it's probably not worth it. > ... This is to say that > > sage: E(5)+E(6) > E(15) + E(15)^4 + E(15)^7 - E(15)^8 > > should always hold true. And I don't yet see (yet) how to ensure this > - in particular, the embedding for prime n's does at least fix the > embedding of all others. No, they don't. All you need is that E(9)^3==E(3), so once you've fixed the image of E(3), the choice of E(9) is only determined up to cube roots of unity. UCF is not finitely generated and not given by finitely many relations, so you can't verify in a finite amount of computation whether arbitrary generator images define a consistent embedding. I'd say just document that it's up to the user to ensure correctness (the alternative, cache all images used and test consistency between those, would allow at least an eventual error but sounds way too onerous to be practical) This is generally a problem and CyclotomicField presently doesn't solve it very nicely either: sage: zC=(CC(-1)^(1/3))^2 sage: Ka.<aa>=CyclotomicField(3,embedding=zC) sage: Kb.<bb>=CyclotomicField(3,embedding=zC^2) sage: Kc.<cc>=CyclotomicField(3,embedding=CC(17)) sage: CC(aa) -0.500000000000000 + 0.866025403784439*I sage: CC(bb) -0.500000000000000 - 0.866025403784439*I sage: CC(Ka(bb)) -0.500000000000000 + 0.866025403784439*I As you see, Ka and Kb apparently have an arbitrary isomorphism between them that is preferred, but of course we can't expect that to be compatible with chosen embeddings. The behaviour with an erroneous embedding is deplorable: The system silently chooses an arbitrary one rather than throw an error or use the specified data and produce the garbage the user deserves: sage: CC(cc) -0.500000000000000 - 0.866025403784439*I -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To post to this group, send email to sage-devel@googlegroups.com. To unsubscribe from this group, send email to sage-devel+unsubscr...@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel?hl=en.
