Function: idealadd
Section: number_fields
C-Name: idealadd
Prototype: GGG
Help: idealadd(nf,x,y): sum of two ideals x and y in the number field
defined by nf.
Doc: sum of the two ideals $x$ and $y$ in the number field $\var{nf}$. The
result is given in HNF.
\bprog
? K = nfinit(x^2 + 1);
? a = idealadd(K, 2, x + 1) \\ ideal generated by 2 and 1+I
%2 =
[2 1]
[0 1]
? pr = idealprimedec(K, 5)[1]; \\ a prime ideal above 5
? idealadd(K, a, pr) \\ coprime, as expected
%4 =
[1 0]
[0 1]
@eprog\noindent
This function cannot be used to add arbitrary $\Z$-modules, since it assumes
that its arguments are ideals:
\bprog
? b = Mat([1,0]~);
? idealadd(K, b, b) \\ only square t_MATs represent ideals
*** idealadd: non-square t_MAT in idealtyp.
? c = [2, 0; 2, 0]; idealadd(K, c, c) \\ non-sense
%6 =
[2 0]
[0 2]
? d = [1, 0; 0, 2]; idealadd(K, d, d) \\ non-sense
%7 =
[1 0]
[0 1]
@eprog\noindent In the last two examples, we get wrong results since the
matrices $c$ and $d$ do not correspond to an ideal: the $\Z$-span of their
columns (as usual interpreted as coordinates with respect to the integer basis
\kbd{K.zk}) is not an $O_K$-module. To add arbitrary $\Z$-modules generated
by the columns of matrices $A$ and $B$, use \kbd{mathnf(concat(A,B))}.