On Fri, May 30, 2008 at 4:33 PM, David Joyner <[EMAIL PROTECTED]> wrote: > > 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.
No, the default is a cyclotomic number field. sage: DirichletGroup(13).base_ring() Cyclotomic Field of order 12 and degree 4 It's possible that Dirichlet characters are too specialized for the tutorial. It made a lot of sense when SAGE = "System for Arithmetic Geometry Experimentation", back when we first put together the tutorial. Now, I'm not so sure. Comments? >> >> 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 > Yes, since DirichletGroup(N) is isomorphic to the dual of (Z/NZ)^*, and dualizing twice gets back to to the original group. Anyway, I would love to have feedback about the topics covered in the tutorial. I think it would make good sense to revisit the choice of topics and "freshen it". Any volunteers or suggestions? -- William --~--~---------~--~----~------------~-------~--~----~ 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 -~----------~----~----~----~------~----~------~--~---
