Karim BELABAS on Fri, 10 Dec 1999 19:39:07 +0100 (MET)

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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 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/