bnrisgalois

Function: bnrisgalois
Section: number_fields
C-Name: bnrisgalois
Prototype: lGGG
Help: bnrisgalois(bnr, gal, H): check whether the class field attached to
 the subgroup H is Galois over the subfield of bnr.nf fixed by the Galois
 group gal, which can be given as output by galoisinit, or as a matrix or a
 vector of matrices as output by bnrgaloismatrix. The ray class field
 attached to bnr need to be Galois, which is not checked.
Doc: check whether the class field attached to the subgroup $H$ is Galois
 over the subfield of \kbd{bnr.nf} fixed by the group \var{gal}, which can be
 given as output by \tet{galoisinit}, or as a matrix or a vector of matrices as
 output by \kbd{bnrgaloismatrix}, the second option being preferable, since it
 saves the recomputation of the matrices.  Note: The function assumes that the
 ray class field attached to bnr is Galois, which is not checked.

 In the following example, we lists the congruence subgroups of subextension of
 degree at most $3$ of the ray class field of conductor $9$ which are Galois
 over the rationals.

 \bprog
 ? K = bnfinit(a^4-3*a^2+253009); B = bnrinit(K,9); G = galoisinit(K);
 ? [H | H<-subgrouplist(B,3), bnrisgalois(B,G,H)];
 time = 160 ms.
 ? M = bnrgaloismatrix(B,G);
 ? [H | H<-subgrouplist(B,3), bnrisgalois(B,M,H)]
 time = 1 ms.
 @eprog
 The second computation is much faster since \kbd{bnrgaloismatrix(B,G)} is
 computed only once.