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D.4.20.1 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:
 
See also: primitive_extra.


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