1Function: rnfidealfactor
2Section: number_fields
3C-Name: rnfidealfactor
4Prototype: GG
5Help: rnfidealfactor(rnf,x): factor the ideal x into
6 prime ideals in the number field nfinit(rnf).
7Doc: factor into prime ideal powers the
8 ideal $x$ in the attached absolute number field $L = \kbd{nfinit}(\var{rnf})$.
9 The output format is similar to the \kbd{factor} function, and the prime
10 ideals are represented in the form output by the \kbd{idealprimedec}
11 function for $L$.
12 \bprog
13 ? rnf = rnfinit(nfinit(y^2+1), x^2-y+1);
14 ? rnfidealfactor(rnf, y+1)  \\ P_2^2
15 %2 =
16 [[2, [0,0,1,0]~, 4, 1, [0,0,0,2;0,0,-2,0;-1,-1,0,0;1,-1,0,0]] 2]
17
18 ? rnfidealfactor(rnf, x) \\ P_2
19 %3 =
20 [[2, [0,0,1,0]~, 4, 1, [0,0,0,2;0,0,-2,0;-1,-1,0,0;1,-1,0,0]] 1]
21
22 ? L = nfinit(rnf);
23 ? id = idealhnf(L, idealhnf(L, 25, (x+1)^2));
24 ? idealfactor(L, id) == rnfidealfactor(rnf, id)
25 %6 = 1
26 @eprog\noindent Note that ideals of the base field $K$ must be explicitly
27 lifted to $L$ via \kbd{rnfidealup} before they can be factored.
28