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