Thomas Obkircher on Sun, 17 Sep 2006 23:25:17 +0200

 Ideal representation in relative extensions

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".

The concrete problem ist that I want to verify whether the Classgroup of
L/Q corresponds to a given representation in L/K.

py=y^2-5;
w5= Mod(y,py);
K = bnfinit(py);

px = x^4+15*x^2+55;
mu = Mod(x,px);
L = bnfinit(px);

p_rel = x^2+(8+(-1+w5)/2);
K_L = rnfinit(K, p_rel);

// ok, now we want the representations of the classgroup generators in
L/K e.g.

g1_rel = rnfidealabstorel(K_L, K.gen[1])
// [[[1, 0]~, [0, -1]~; [0, 0]~, [1, 0]~], [[3, 0; 0, 3], [1, 0; 0, 1]]]

g2_rel = rnfidealabstorel(K_L, K.gen[2])
//  [[[1, 0]~, [-1, -1]~; [0, 0]~, [1, 0]~], [[2, 0; 0, 2], [1, 0; 0,
1]]]

g3_rel = rnfidealabstorel(K_L, K.gen[1]*K.gen[2])
// [[[1, 0]~, [-3, 2]~; [0, 0]~, [1, 0]~], [[6, 0; 0, 6], [1, 0; 0, 1]]]

...
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?

Thanks in advance for some explanatory words...

Thomas