Re: bnrinit

[Karim BELABAS <Karim.Belabas@math.u-psud.fr>]

On 21 Oct 2003, Markus Endres wrote:
> for example I use
>
> ? bnf = bnfinit(x^4 + 2*x^3 + 25*x^2 + 24*x + 111);
>
> ? ideal = [3,0,1,0;0,3,0,1;0,0,1,0;0,0,0,1];
>
> ? rcgp = bnrinit(bnf,ideal,1);
>
> than I get
>
> ? rcgp.clgp
> %1 = [1, [], []]
> ?
>
> now, what's the meaning of the empty vectors in rcgp.clgp. I know the
> structure is like [cardinality, vector of cyclic components,
> generators]. but I would expect something like [1, [0], [0]].

This is not specific to bnrinit, but applies to all abelian group structures
in PARI. Cyclic components are written (uniquely) as

     Z/(d_1) \oplus Z/(d_2) \oplus ... \oplus Z/(d_r),

where d_1 | d_2 | ... | d_r and d_1 > 1. If cardinality is 1, then r = 0,
and the direct sum is empty. Hence the empty vectors.

Cheers,

    Karim.
-- 
Karim Belabas                     Tel: (+33) (0)1 69 15 57 48
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Université Paris-Sud              http://www.math.u-psud.fr/~belabas/
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