Function: rnfidealfactor
Section: number_fields
C-Name: rnfidealfactor
Prototype: GG
Help: rnfidealfactor(rnf,x): factor the ideal x into
 prime ideals in the number field nfinit(rnf).
Doc: factor into prime ideal powers the
 ideal $x$ in the attached absolute number field $L = \kbd{nfinit}(\var{rnf})$.
 The output format is similar to the \kbd{factor} function, and the prime
 ideals are represented in the form output by the \kbd{idealprimedec}
 function for $L$.
 \bprog
 ? rnf = rnfinit(nfinit(y^2+1), x^2-y+1);
 ? rnfidealfactor(rnf, y+1)  \\ P_2^2
 %2 =
 [[2, [0,0,1,0]~, 4, 1, [0,0,0,2;0,0,-2,0;-1,-1,0,0;1,-1,0,0]] 2]

 ? rnfidealfactor(rnf, x) \\ P_2
 %3 =
 [[2, [0,0,1,0]~, 4, 1, [0,0,0,2;0,0,-2,0;-1,-1,0,0;1,-1,0,0]] 1]

 ? L = nfinit(rnf);
 ? id = idealhnf(L, idealhnf(L, 25, (x+1)^2));
 ? idealfactor(L, id) == rnfidealfactor(rnf, id)
 %6 = 1
 @eprog\noindent Note that ideals of the base field $K$ must be explicitly
 lifted to $L$ via \kbd{rnfidealup} before they can be factored.