#17317: Add unit_group() method to IntegerModRing
-------------------------------------+-------------------------------------
Reporter: pbruin | Owner:
Type: enhancement | Status: needs_work
Priority: major | Milestone: sage-6.4
Component: number theory | Resolution:
Keywords: unit group | Merged in:
Authors: Peter Bruin | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
u/pbruin/17317-IntegerModRing_unit_group|
7f38953bdbc38709a218533934d759abc5325f76
Dependencies: #17315 | Stopgaps:
-------------------------------------+-------------------------------------
Description changed by pbruin:
Old description:
> This ticket implements a `unit_group()` method for '''Z'''/''n'''''Z'''.
> Example:
> {{{
> sage: A = Zmod(24)
> sage: G = A.unit_group(); G
> Multiplicative Abelian group isomorphic to C2 x C2 x C2
> sage: G.gens_orders()
> (2, 2, 2)
> sage: G.gens_values()
> (7, 13, 17)
> }}}
>
> At the moment, there is no new class for such groups; it uses
> `AbelianGroupWithValues`. However, this could easily be changed in the
> future if required.
>
> The `unit_group()` method admits an optional `algorithm` argument
> (default: `'sage'`). This can be set to `'pari'` to use PARI's
> `znstar()` function (see #17315). This gives different generators in
> general.
>
> The ticket also makes the set of generators for ''n'' = 1 be the empty
> set, and makes Dirichlet characters modulo 1 and 2 print as
> {{{
> sage: DirichletGroup(1)[0]
> Dirichlet character modulo 1 of conductor 1
> sage: DirichletGroup(2)[0]
> Dirichlet character modulo 2 of conductor 1
> }}}
> i.e. without the `mapping ...` appearing in
> {{{
> sage: DirichletGroup(3)[1]
> Dirichlet character modulo 3 of conductor 1 mapping 2 |--> -1
> }}}
>
> See also:
>
> - #7234
> - https://groups.google.com/forum/#!topic/sage-devel/T5A_gwrtZT0
New description:
This ticket implements a `unit_group()` method for '''Z'''/''n'''''Z'''.
Example:
{{{
sage: A = Zmod(24)
sage: G = A.unit_group(); G
Multiplicative Abelian group isomorphic to C2 x C2 x C2
sage: G.gens_orders()
(2, 2, 2)
sage: G.gens_values()
(7, 13, 17)
}}}
At the moment, there is no new class for such groups; it uses
`AbelianGroupWithValues`. However, this could easily be changed in the
future if required.
The `unit_group()` method admits an optional `algorithm` argument
(default: `'sage'`). This can be set to `'pari'` to use PARI's `znstar()`
function (see #17315). This gives different generators in general.
See also:
- #7234
- https://groups.google.com/forum/#!topic/sage-devel/T5A_gwrtZT0
--
--
Ticket URL: <http://trac.sagemath.org/ticket/17317#comment:8>
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