Karim BELABAS on Fri, 10 Dec 1999 19:39:07 +0100 (MET) |
[Date Prev] [Date Next] [Thread Prev] [Thread Next] [Date Index] [Thread Index]
Re: relative number fields |
[Annegret Weng:] > I have a problem concerning pseudo-bases. I do not understand what the > ideal list means. > > For example: > I have the number field K=Q(sqrt(3)) generated under gp > by nf=nfinit(x^2-3,0). Now I define an extension L=K(sqrt(a)) > generated by the element sqrt(a) where a=7+2*sqrt(3). Since Q(sqrt(3)) has > class number one there must be a relative integral basis for L over K. > And in fact I get such a basis by using the function rnfbasis. > > My questions: > Can I get the relative integral basis only by using the > function rnfinit? Not unless you were lucky and all ideals in the pseudo basis are equal to Z_K (rnfinit doesn't even try to make it that way). > Can I use the information from rnf[7] which is > for this example [[Mod(1, y^2 - 3), Mod(1, y^2 - 3)*x + Mod(1, y^2 - 3)], > [[1, 0; 0, 1], [1, 1/2; 0, 1/2]]] ? Let's call [[a,b], [I,J]] the full pseudo-basis, where a,b in L, I,J (fractionnal) ideals of K. This means that O_L = a I + b J > (Note that the elements [Mod(1, y^2 -3), Mod(1, y^2 - 3)*x + Mod(1, y^2 - 3)] over O_K generate an order, but > not the maximal order O_L.) Indeed since I and J are not both equal to Z_K. In this case, I = Z_K, so it's a matter of finding a generator of J. Well, ? bnf = bnfinit(nf); ? u = bnfisprincipal(bnf, [1, 1/2; 0, 1/2]) %3 = [[]~, [1/2, 1/2]~, 121] ? u = nfalgtobasis(nf,u[2]) %4 = Mod(1/2*y + 1/2, y^2 - 3) and [a, b * u] form a relative basis. In general, that's exactly what rnfbasis does: reduce to Steinitz form (all the ideals equal to Z_K but the last one), then check whether the Steinitz class is trivial, and if so output a generator, otherwise find a 2-elt representative for that ideal. Since one has to compute generators of principal ideals, I see no general way to avoid computing a full bnf (for very easy fields such as the one above, you can do without the bnf, but then there's no problem computing it in the first place, so...) Karim. __ Karim Belabas email: Karim.Belabas@math.u-psud.fr Dep. de Mathematiques, Bat. 425 Universite Paris-Sud Tel: (00 33) 1 69 15 57 48 F-91405 Orsay (France) Fax: (00 33) 1 69 15 60 19 -- PARI/GP Home Page: http://hasse.mathematik.tu-muenchen.de/ntsw/pari/