Karim Belabas on Mon, 18 Sep 2006 11:42:29 +0200 |
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Re: Ideal representation in relative extensions |
* Thomas Obkircher [2006-09-17 23:25]: > 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); 1) I notice (and will use that fact later) that px = K_L.pol 2) There is a distinct potential for confusion here since x is the main variable of p_rel representing the element (x mod p_rel) in L ~ K[x] / p_rel, _and_ of px representing the element (x mod px) in L = Q[x] / px. Fortunately, from the 3rd component of K_L[11] ( = 0 ), these two elements are in fact one and the same (via the above isomorphism), so we should be fine ! > // ok, now we want the representations of the classgroup generators in > L/K e.g. > > g1_rel = rnfidealabstorel(K_L, K.gen[1]) You probably mean L.gen[1]. (O_K is principal, so K.gen[1] doesn't exist.) Then the correct command (see ??rnfidealabstorel) would be g1_rel = rnfidealabstorel(K_L, L.zk * L.gen[1]) and I indeed get: > // [[[1, 0]~, [0, -1]~; [0, 0]~, [1, 0]~], [[3, 0; 0, 3], [1, 0; 0, 1]]] 1) K.zk says that the chosen integer basis for the base field is [1, (w5-1)/2 ]. 2) K_L[7] says that the relative integer basis is O_K + x O_K 3) The pseudo-matrix above has colums a = [[1, 0]~, [0, 0]~]~ = 1 b = [[0, -1]~, [1, 0]~]~ = -(w5-1)/2 + x So the above "relative ideal" really is 3 O_K + b O_K > 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]]] You probably want to introduce idealmul(L, L.gen[1], L.gen[2]) here [ not ordinary matrix multiplication ! ] > ... > 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 In GP, that would be A_2 = [[-3, 0; (y-1)/2, 1], [idealhnf(K,1), idealhnf(K,1)]] Note that this is actually 3 O_K + mu O_K, since (y-1)/2 is integral. Also N_{L/K}(mu) = -(8+(-1+w5)/2), and the absolute norm of the latter is 55, which is coprime to 3. [ norm(Mod(x,px) --> 55 confirms that computation ] Assuming A_2 is indeed an ideal (i.e an O_L-module), it must be equal to O_L ! Let us check this assumption: what is mu^2, well -(8+(-1+w5)/2) obviously. Unfortunately, the latter does not belong to 3 O_K, so A_2 is not an O_L-ideal and we are in trouble. Further notes: The documentation of rnfidealreltoabs introduces the following helper routine: idealgentoHNF(nf, y) = mathnf( Mat( nfalgtobasis(nf, y) ) ); ? idealgentoHNF(L, rnfidealreltoabs(K_L, A_2)) %5 = [3 0 0 0] [0 3 0 0] [0 0 1 0] [0 0 0 1] which, as expected and according to nfisideal(L,%), is still not an ideal. > 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? Yes, they are expressed in terms of the pseudo matrix which is the first component of the relative Z_L basis given by K_L[7]. > How can I see the correspondance to the mentioned ideals? > > Thanks in advance for some explanatory words... Hope this helps (and does not contain too many typos). Cheers, K.B. -- Karim Belabas Tel: (+33) (0)5 40 00 26 17 Universite Bordeaux 1 Fax: (+33) (0)5 40 00 69 50 351, cours de la Liberation http://www.math.u-bordeaux.fr/~belabas/ F-33405 Talence (France) http://pari.math.u-bordeaux.fr/ [PARI/GP] `