Function: idealramgroups
Section: number_fields
C-Name: idealramgroups
Prototype: GGG
Help: idealramgroups(nf,gal,pr): let pr be a prime ideal pr in prid format, in
 the Galois group gal 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 $nf$ and assume $K/\Q$ be a
 Galois extension with Galois group $G$ given \kbd{gal=galoisinit(nf)},
 and that $pr$ is 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 ramification group of
 $\goth{P}$.

 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 $e=1$.

 \bprog
 ? nf=nfinit(x^6+108);
 ? gal=galoisinit(nf);
 ? pr=idealprimedec(nf,2)[1];
 ? iso=idealramgroups(nf,gal,pr)[2]
 %4 = [[Vecsmall([2, 3, 1, 5, 6, 4])], Vecsmall([3])]
 ? nfdisc(galoisfixedfield(gal,iso,1))
 %5 = -3
 @eprog\noindent The field fixed by the inertia group of $2$ is not ramified at
 $2$.
