#17317: Add unit_group() method to IntegerModRing
-----------------------------+-------------------------------
Reporter: pbruin | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.4
Component: number theory | Keywords: unit group
Merged in: | Authors: Peter Bruin
Reviewers: | Report Upstream: N/A
Work issues: | Branch:
Commit: | Dependencies: #17315
Stopgaps: |
-----------------------------+-------------------------------
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
--
Ticket URL: <http://trac.sagemath.org/ticket/17317>
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