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.