Bill Allombert on Mon, 18 Sep 2006 00:50:59 +0200 |
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Re: Ideal representation in relative extensions |
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.