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