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.
__
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
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