Singular

D.4.23.1 primitive

Procedure from library `primitiv.lib` (see primitiv_lib).

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.

Example:
 ```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. ```