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From: Jude on 21 Jun 2010 06:44
Hi~

We know that the ideal class group of a number field is finitely
generated abelain group and so it is isomorphic to the product of
finite cyclic group.

Then, for each cyclic component, can we choose a prime ideal whose
class is the generator of the cyclic component?
From: Arturo Magidin on 21 Jun 2010 10:30
On Jun 21, 5:44 am, Jude <classnu...(a)gmail.com> wrote:
> Hi~
>
> We know that the ideal class group of a number field is finitely
> generated abelain group and so it is isomorphic to the product of
> finite cyclic group.

Well, we know the ideal class group of a number field is a *finite*
abelian group (by Finiteness of the Class Number); that's why you get
it as a product/sum of *finite* cyclic groups. Finite generation would
only get you that the group is a product of cyclic groups, possibly
infinite. This is what happens, for example, when we look at other
situations like the function field case, where the ideal class group
is still finitely generated, but is not necessarily finite.

> Then, for each cyclic component, can we choose a prime ideal whose
> class is the generator of the cyclic component?

Of course. You don't even need the Axiom of Choice to do it. You have
finitely many equivalence classes, you can always pick a
representative from each.

Or did you mean something "constructive" in some sense?

--
Arturo Magidin
From: Masato Takahashi on 27 Jun 2010 00:50
On Jun 21, 11:30 pm, Arturo Magidin <magi...(a)member.ams.org> wrote:
> On Jun 21, 5:44 am, Jude <classnu...(a)gmail.com> wrote:
>
> > Hi~
>
> > We know that the ideal class group of a number field is finitely
> > generated abelain group and so it is isomorphic to the product of
> > finite cyclic group.
>
> Well, we know the ideal class group of a number field is a *finite*
> abelian group (by Finiteness of the Class Number); that's why you get
> it as a product/sum of *finite* cyclic groups. Finite generation would
> only get you that the group is a product of cyclic groups, possibly
> infinite. This is what happens, for example, when we look at other
> situations like the function field case, where the ideal class group
> is still finitely generated, but is not necessarily finite.
>
> > Then, for each cyclic component, can we choose a prime ideal whose
> > class is the generator of the cyclic component?
>
> Of course. You don't even need the Axiom of Choice to do it. You have
> finitely many equivalence classes, you can always pick a
> representative from each.
>
> Or did you mean something "constructive" in some sense?
>
> --
> Arturo Magidin

Does each ideal class contains a prime ideal?
From: Arturo Magidin on 27 Jun 2010 16:02
On Jun 26, 11:50 pm, Masato Takahashi <genkim...(a)gmail.com> wrote:
> On Jun 21, 11:30 pm, Arturo Magidin <magi...(a)member.ams.org> wrote:
>
>
>
> > On Jun 21, 5:44 am, Jude <classnu...(a)gmail.com> wrote:
>
> > > Hi~
>
> > > We know that the ideal class group of a number field is finitely
> > > generated abelain group and so it is isomorphic to the product of
> > > finite cyclic group.
>
> > Well, we know the ideal class group of a number field is a *finite*
> > abelian group (by Finiteness of the Class Number); that's why you get
> > it as a product/sum of *finite* cyclic groups. Finite generation would
> > only get you that the group is a product of cyclic groups, possibly
> > infinite. This is what happens, for example, when we look at other
> > situations like the function field case, where the ideal class group
> > is still finitely generated, but is not necessarily finite.
>
> > > Then, for each cyclic component, can we choose a prime ideal whose
> > > class is the generator of the cyclic component?
>
> > Of course. You don't even need the Axiom of Choice to do it. You have
> > finitely many equivalence classes, you can always pick a
> > representative from each.
>
> > Or did you mean something "constructive" in some sense?

> Does each ideal class contains a prime ideal?

I missed the "prime" the first time around. You can pick primes whose
classes generate the ideal class group, but I do not know if you can
pick primes whose classes are the generators of the cyclic factors.

--
Arturo Magidin
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