Function: idealmul
Section: number_fields
C-Name: idealmul0
Prototype: GGGD0,L,
Help: idealmul(nf,x,y,{flag=0}): product of the two ideals x and y in the
number field nf. If (optional) flag is non-nul, reduce the result.
Description:
(gen, gen, gen, ?0):gen idealmul($1, $2, $3)
(gen, gen, gen, 1):gen idealmulred($1, $2, $3)
(gen, gen, gen, #small):gen $"invalid flag in idealmul"
(gen, gen, gen, small):gen idealmul0($1, $2, $3, $4)
Doc: ideal multiplication of the ideals $x$ and $y$ in the number field
\var{nf}; the result is the ideal product in HNF. If either $x$ or $y$
are extended ideals\sidx{ideal (extended)}, their principal part is suitably
updated: i.e. multiplying $[I,t]$, $[J,u]$ yields $[IJ, tu]$; multiplying
$I$ and $[J, u]$ yields $[IJ, u]$.
\bprog
? nf = nfinit(x^2 + 1);
? idealmul(nf, 2, x+1)
%2 =
[4 2]
[0 2]
? idealmul(nf, [2, x], x+1) \\ extended ideal * ideal
%3 = [[4, 2; 0, 2], x]
? idealmul(nf, [2, x], [x+1, x]) \\ two extended ideals
%4 = [[4, 2; 0, 2], [-1, 0]~]
@eprog\noindent
If $\fl$ is non-zero, reduce the result using \kbd{idealred}.
Variant:
\noindent See also
\fun{GEN}{idealmul}{GEN nf, GEN x, GEN y} ($\fl=0$) and
\fun{GEN}{idealmulred}{GEN nf, GEN x, GEN y} ($\fl\neq0$).