Unit group

[David Kohel]

Hi,

It is a general design question whether the unit group of a number
field
K should be K^* or O_K^*, since the former is also an important group
in number theory. I would argue for the latter, but Magma doesn't
agree
(or is locked into its prior design decision), and Sage has followed
in
this path.

Opinions? Comments?

--David

[Alex Ghitza]

Hi David,

Question? I must be misunderstanding what you are saying, because it
seems to me that in Sage the unit group is O_K^*:

sage: K.<a> = QuadraticField(-3)
sage: UK = K.unit_group(); UK
Unit group with structure C6 of Number Field in a with
defining polynomial x^2 + 3
sage: UK.gens()
[-1/2*a + 1/2]

Best,
Alex

--
Alex Ghitza -- http://aghitza.org/
Lecturer in Mathematics -- The University of Melbourne -- Australia

[William Stein]

I'm confused by what you mean by "former" and "later" above, since it
seems you got things
mixed up. Just for clarification, in Sage "the unit group of a number
field K" is O_K^*.

sage: K.<a> = NumberField(x^3 + x+1)
sage: K.unit_group()
Unit group with structure C2 x Z of Number Field in a with defining
polynomial x^3 + x + 1

which is the same as in Magma:

sage: Km = magma(K)
sage: Km.UnitGroup()
Abelian Group isomorphic to Z/2 + Z
Defined on 2 generators
Relations:
2*$.1 = 0

http://www.sagemath.org/doc/reference/sage/rings/number_field/unit_group.html

It's *definitely* indisputably the case that any paper I've ever seen
in number theory that used the phrase "unit group of a number
field..." means O_K^*. However, obviously, if the paper wrote "K^*",
that would mean the nonzero elements of K, which may be a useful
object to support as well.

If Sage didn't already have K.unit_group() mean "O_K^*" for several
years now, I would be more interesting in discussing this. As is, it
would be a little scary to change this and hence break code.

What about the following compromise:

sage: K.<a> = NumberField(x^3 + x+1)
sage: K.unit_group()
Unit group with structure C2 x Z of Number Field in a with defining
polynomial x^3 + x + 1
sage: K.multiplicative_group()
Multiplicative group of nonzero elements of Number Field in a with
defining polynomial x^3 + x + 1

Moreover, the docstrings for unit_group and multiplicative_group would
both immediately explain the "situation".

-- William

> Opinions? Comments?
>
> --David
>
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--
William Stein
Professor of Mathematics
University of Washington
http://wstein.org

[David R. Kohel]

My error: Correct to: "I would argue for the former.". If K.unit_group()
returns O_K^*, then we no longer have any means of constructing K^*.
How do we create det: GL_n(K) -> K^*? How do we compute group cohomology
of K^*? --David

[Marco Streng]

2010/6/15 William Stein <wst...@gmail.com>:

> What about the following compromise:
>
> sage: K.<a> = NumberField(x^3 + x+1)
> sage: K.unit_group()
> Unit group with structure C2 x Z of Number Field in a with defining
> polynomial x^3 + x + 1

For this to be absolutely clear and mathematically correct, replace
"of" by "of the maximal order of" (and do the same throughout the
documentation).

Shouldn't there also be a function unit_group() for orders? Then we
could have K.unit_group() and K.maximal_order().unit_group() give the
same output.

ps. I would have also said that K.unit_group() should return K^* (and
was surprised when I found out for the first time that it didn't), but
agree with WiIliam's argument about not breaking code.

[William Stein]

On Tue, Jun 15, 2010 at 12:39 AM, David R. Kohel <ko...@iml.univ-mrs.fr> wrote:
> My error: Correct to: "I would argue for the former.".  If K.unit_group()
> returns O_K^*, then we no longer have any means of constructing K^*.
> How do we create det: GL_n(K) -> K^*? How do we compute group cohomology
> of K^*? --David

I explained in detail in my post how to create K^*. Please reread it.

-- William

[Nick Alexander]

I think it would be more consistent to move things pertaining to the
ring of integers (or more generally an order) of a number field into
the objects we have for that purpose. So:

sage: K = NumberField(x^2 - 2, 'a')
sage: K.unit_group()
K^*
sage: K.maximal_order().unit_group()
O_K^*

But that may not be possible due to backward compatibility.

Nick