Re: Ideal representation in relative extensions

[Bill Allombert <allomber@math.u-bordeaux.fr>]

On Sun, Sep 17, 2006 at 11:17:04PM +0200, Thomas Obkircher wrote:
> Hi,
> 
> I'm still quite new to PARI and I'm a little bit stuck at this moment.
> I'm dealing with ideals in relative extensions, I read the relevant
> parts in the Manual as well as the other mailing list entries concerning
> relative extensions, but I still don't understand the notion of a
> "relative pseudo basis".
> 
> these relative representations should correspond somehow to the
> following ideals, where O_K is the ring of integers of K:
> A_2 = (-3+      (-1+\sqrt(5))/2 \mu) O_K + \mu O_K
> A_3 = (-2+ (1 + (-1+\sqrt(5))/2 \mu) O_K + \mu O_K
> A_4 = (-6+ (3 + (-1+\sqrt(5))/2 \mu) O_K + \mu O_K
> 
> Accordung the manual relative ideals are given by a "relative pseudo
> matrix" which is the sum over the ideals multiplied by the according
> columns of the first matrix, whose entries are given in a O_K Basis.

> But what does this exactly mean? We have A a + B b , where A,B are
> Ideals and a,b are VECTORS. Is this a representation in the Z_L basis of
> the extension? How can I see the correspondance to the mentioned ideals?

a, b are vectors representing elements of L over some basis of L over K, 
though I am not quite sure whether the basis is K_L[7][1] or
[1,x,...,x^(n-1)] but this is the same here.  This point of the
documentation should be clarified. Maybe it was the missing clue ?

Here how that plays out:

A and B are ideals of Z_K. where a and b are elements of L represented
by vector over Z_K.

For example for A2:
A_2 = (-3+      (-1+\sqrt(5))/2 \mu) O_K + \mu O_K
we have A=O_K, B=O_K, 
a= (-3+      (-1+\sqrt(5))/2 \mu) 
b=\mu
The fact that O_K is principal implay that it is always 
possible to choose A=O_K, B=O_K.

Before going further, let us note the basis of O_K over Z:
? K.zk
%9 = [1, 1/2*y - 1/2]
Thus [1, (sqrt(5)-1)/2]

now what mean the PARI result:
I=[[[1, 0]~, [0, -1]~; [0, 0]~, [1, 0]~], [[3, 0; 0, 3], [1, 0; 0, 1]]]
     a_1       b_1     a_2       b_2            A             B

We have:
A=3O_K B=O_K
a=[[1, 0]~,[0, 0]~]~ = [1,0]=1
b=[[0, -1]~,[1, 0]~] = [(1-sqrt(5))/2,1]=(1-sqrt(5))/2+mu
So
I=3O_K+((1-sqrt(5))/2+mu)O_K

(I hope I did not make any mistake).

Cheers,
Bill.