Function: rnfidealprimedec
Section: number_fields
C-Name: rnfidealprimedec
Prototype: GG
Help: rnfidealprimedec(rnf,pr): return prime ideal decomposition of the maximal
 ideal pr of K in L/K; pr is also allowed to be a prime number p, in which
 case return a pair of vectors [SK,SL], where SK contains the primes of K
 above p and SL[i] is the vector of primes of L above SK[i].
Doc: let \var{rnf} be a relative number
 field extension $L/K$ as output by \kbd{rnfinit}, and \var{pr} a maximal
 ideal of $K$ (\var{prid}), this function completes the \var{rnf}
 with a \var{nf} structure attached to $L$ (see \secref{se:rnfinit})
 and returns the vector $S$ of prime ideals of $\Z_{L}$ above \var{pr}.
 \bprog
 ? K = nfinit(y^2+1); rnf = rnfinit(K, x^3+y+1);
 ? pr = idealprimedec(K, 2)[1];
 ? S = rnfidealprimedec(rnf, pr);
 ? #S
 %4 = 1
 @eprog\noindent The relative ramification indices and residue degrees
 can be obtained as \kbd{PR.e / pr.e} and \kbd{PR.f / PR.f}, if \kbd{PR}
 is an element of $S$.

 The argument \var{pr} is also allowed to be a prime number $p$, in which
 case the function returns a pair of vectors \kbd{[SK,SL]}, where \kbd{SK}
 contains the primes of $K$ above $p$ and \kbd{SL}$[i]$ is the vector of primes
 of $L$ above \kbd{SK}$[i]$.
 \bprog
 ? [SK,SL] = rnfidealprimedec(rnf, 5);
 ? [#SK, vector(#SL,i,#SL[i])]
 %6 = [2, [2, 2]]
 @eprog

