D.4.28.1 primitive

Usage:

primitive(i); i ideal

Assume:

i is given by generators m[1],...,m[n] such that for j=1,...,n
- m[j] is a polynomial in k[x(1),...,x(j)]
- m[j](a[1],...,a[j-1],x(j)) is the minimal polynomial for a[j] over k(a[1],...,a[j-1])
(k the ground field of the current basering and x(1),...,x(n) the ring variables).

Return:

ideal j in k[x(n)] with
- j[1] a minimal polynomial for a primitive element b of k(a[1],...,a[n]) over k,
- j[2],...,j[n+1] polynomials in k[x(n)] such that j[i+1](b)=a[i] for i=1,...,n.

Note:

the number of variables in the basering has to be exactly n, the number of given generators (i.e., minimal polynomials).
If the ground field k has only a few elements it may happen that no linear combination of a[1],...,a[n] is a primitive element. In this case primitive(i) returns the zero ideal, and one should use primitive_extra(i) instead.

LIB "primitiv.lib";
ring exring=0,(x,y),dp;
ideal i=x2+1,y2-x;                  // compute Q(i,i^(1/2))=:L
ideal j=primitive(i);
j[1];                               // L=Q(a) with a=(-1)^(1/4)
==> y4+1
j[2];                               // i=a^2
==> y2
j[3];                               // i^(1/2)=a
==> y
// the 2nd element was already primitive!
j=primitive(ideal(x2-2,y2-3));      // compute Q(sqrt(2),sqrt(3))
j[1];
==> y4-10y2+1
j[2];
==> 1/2y3-9/2y
j[3];
==> -1/2y3+11/2y
// no element was primitive -- the calculation of primitive elements
// is based on a random choice.