Function: idealramgroups
Section: number_fields
C-Name: idealramgroups
Prototype: GGG
Help: idealramgroups(nf,gal,pr): let pr be a prime ideal in prid format, and
gal the Galois group of the number field nf, return a vector g such that g[1]
is the decomposition group of pr, g[2] is the inertia group, g[i] is the
(i-2)th ramification group of pr, all trivial subgroups being omitted.
Doc: Let $K$ be the number field defined by \var{nf} and assume that $K/\Q$ is
Galois with Galois group $G$ given by \kbd{gal=galoisinit(nf)}.
Let \var{pr} be the prime ideal $\goth{P}$ in prid format.
This function returns a vector $g$ of subgroups of \kbd{gal}
as follow:
\item \kbd{g[1]} is the decomposition group of $\goth{P}$,
\item \kbd{g[2]} is $G_0(\goth{P})$, the inertia group of $\goth{P}$,
and for $i\geq 2$,
\item \kbd{g[i]} is $G_{i-2}(\goth{P})$, the $i-2$-th
\idx{ramification group} of $\goth{P}$.
\noindent The length of $g$ is the number of non-trivial groups in the
sequence, thus is $0$ if $e=1$ and $f=1$, and $1$ if $f>1$ and $e=1$.
The following function computes the cardinality of a subgroup of $G$,
as given by the components of $g$:
\bprog
card(H) =my(o=H[2]); prod(i=1,#o,o[i]);
@eprog
\bprog
? nf=nfinit(x^6+3); gal=galoisinit(nf); pr=idealprimedec(nf,3)[1];
? g = idealramgroups(nf, gal, pr);
? apply(card,g)
%3 = [6, 6, 3, 3, 3] \\ cardinalities of the G_i
@eprog
\bprog
? nf=nfinit(x^6+108); gal=galoisinit(nf); pr=idealprimedec(nf,2)[1];
? iso=idealramgroups(nf,gal,pr)[2]
%5 = [[Vecsmall([2, 3, 1, 5, 6, 4])], Vecsmall([3])]
? nfdisc(galoisfixedfield(gal,iso,1))
%6 = -3
@eprog\noindent The field fixed by the inertia group of $2$ is not ramified at
$2$.