Karim Belabas on Mon, 03 Apr 2017 08:47:04 +0200

 Re: Reuse of data in rnfkummer/computing order of galois elements without it

```* Karim Belabas [2017-04-03 08:07]:
> > 2: Is there a way to determine the maximal order of an element of the
> > Galois group of the resulting field without computing the field via
> > rnfkummer? I know about bnrgaloismat, but it doesn't seem as though
> > the resulting representation of the galois group can be used to find
> > the order of elements easily.
>
> By "Galois group" I understand the Galois group of (class field) over K.
> (If you meant Gal(class field/Q), modify the following with the maximal
> order allowed in the quotient.)
>
> I don't see a fast if-and-onlu-if algorithm. There's a quick early abort
> though : pick prime ideals in the base field not dividing the conductor
> and compute their order in the ray class group (see below); if this
> order is too large, stop. Under GRH, a suitable effective Chebotarev
> bound would turn this into a fast rigorous algorithm. In practice, if
> the test does not find elements of large order, it means they (probably)
> don't exist and you can go on with rnfkummer...
>
> N.B. If
>   C = bnrinit( bnfinit(K), f );
>   pr = idealprimedec(C, p)[1];
> then
>   charorder(C, bnrisprincipal(C, pr, 0))
> is the order of cl(pr) in Cl_f(K)

I was confused (never think about math before breakfast...).

I guess what you really have as input is a pair (f,H) attached to a
class field L/K, not necessarily a ray class field as in my answer.
Where f a divisor and H < Cl_f(K) is a congruence subgroup.

Then
1) the exponent of the quotient G = Cl_f(K) / H is the largest order for
elements in Gal(L/K);
2) if H is given by a matrix whose columns describe its generators on C.gen
(e.g. a left-divisor of matdiagonal(C.cyc) from subgrouplist), then matsnf(H)
gives the structure of G as a product of cyclic groups (SNF).

Cheers,

K.B.
--
Karim Belabas, IMB (UMR 5251)  Tel: (+33) (0)5 40 00 26 17
Universite de Bordeaux         Fax: (+33) (0)5 40 00 21 23
351, cours de la Liberation    http://www.math.u-bordeaux.fr/~kbelabas/
F-33405 Talence (France)       http://pari.math.u-bordeaux.fr/  [PARI/GP]
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