On May 30, 5:17 pm, "William Stein" <[EMAIL PROTECTED]> wrote:
> 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?
Well, as I said, I'm trying to edit it. This will take several passes:
copy-editing (rewording, making the typesetting look better, etc.),
and then when I've done a big enough chunk, or when I see something
that seems out of place, thinking about re-organizing things. I don't
know if I have enough vision of the whole project to know what should
be included in the tutorial if it isn't already there. Of what's
there, I have the following preliminary comments, and I'll have more
as I work my way through:
1. it's a little odd to start the guided tour with a table of
"Arithmetical binary operator precedence". Sort of like starting a
tour of the university with the physical plant. This sort of thing
should be mentioned, probably with a link, but putting it first is
strange.
2. In my edited version, I've moved the stuff about p-adic numbers out
of what is essentially the second page of the tour ("Basic, and not-so-
basic, Rings", now just called "Rings") and put it in a new subsection
of the "Number theory" section: it seemed odd to talk about p-adics
before polynomials, for example.
John
>
> -- William
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