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