On Fri, May 30, 2008 at 5:27 PM, John H Palmieri <[EMAIL PROTECTED]> wrote: > > I'm working on rewriting the tutorial. Section 2.5.1 of the tutorial > is about Dirichlet characters, and I have some questions: > > The introductory sentence discusses "Dirichlet characters", and then > the first example uses "DirichletGroup". What is the connection? > (I'm thinking that for fixed n and R, the set of Dirichlet characters > forms a group, and that's what's being computed here. Is that right? > If so, in the notation for elements of the group, what function does > an element like [1,zeta6] correspond to?)
As you suspected, the set of all Dirichlet characters mod N form a group called the DirichletGroup. The notation [1,zeta6] might be distracting for a beginner, as it is what I would call the SAGE internal representation for a fixed generator chi of DirichletGroup(21). The ordering in which the group elements are listed is (I think) as powers of the generator. For example, the 11th element is the 11th power: sage: G = DirichletGroup(21) sage: len(G) 12 sage: chi1 = G.gens()[1] sage: chi1 [1, zeta6] sage: chi2 = G[10] sage: chi2 [1, -zeta6 + 1] sage: chi2(19) zeta6 sage: chi1(19)^11 zeta6 sage: chi1^11 [1, -zeta6 + 1] sage: chi1^11 == chi2 True > > The introductory sentence also discusses "some ring" R. Is R actually > the complex numbers here? Can we add a parenthetical remark like > "(Often, R is the complex numbers.)"? Yes this sounds good to me. > > The introductory sentence discusses a map with domain (Z/NZ)*. A bit > later, a Galois group involving zeta_n is mentioned. Presumably n = > N? No. The group order of DirichletGroup(N) is the Euler totient \phi(N), so with N=21, n=12: sage: euler_phi(21) 12 > > Overall, this section is a bit challenging for something in a > tutorial, especially in such an early section. I'm hoping that if I > understand the answers to these questions, I can make it gentle enough > to keep, but it might have to be removed, or at least moved back. Thank you very much for working on this! > > > > > --~--~---------~--~----~------------~-------~--~----~ 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://www.sagemath.org -~----------~----~----~----~------~----~------~--~---
