Function: idealfactor
Section: number_fields
C-Name: gpidealfactor
Prototype: GGDG
Help: idealfactor(nf,x,{lim}): factorization of the ideal x into prime ideals
in the number field nf. If lim is set return partial factorization, using
primes < lim.
Doc: factors into prime ideal powers the ideal $x$ in the number field
$\var{nf}$. 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. If \var{lim} is set, return partial
factorization, including only prime ideals above rational primes
$< \var{lim}$.
\bprog
? nf = nfinit(x^3-2);
? idealfactor(nf, x) \\ a prime ideal above 2
%2 =
[[2, [0, 1, 0]~, 3, 1, ...] 1]
? A = idealhnf(nf, 6*x, 4+2*x+x^2)
%3 =
[6 0 4]
[0 6 2]
[0 0 1]
? idealfactor(nf, A)
%4 =
[[2, [0, 1, 0]~, 3, 1, ...] 2]
[[3, [1, 1, 0]~, 3, 1, ...] 2]
? idealfactor(nf, A, 3) \\ restrict to primes above p < 3
%5 =
[[2, [0, 1, 0]~, 3, 1, ...] 2]
@eprog
Variant: This function should only be used by the \kbd{gp} interface. Use
directly \fun{GEN}{idealfactor}{GEN x} or
\fun{GEN}{idealfactor_limit}{GEN x, ulong lim}.