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D.4.20.2 primitive_extra

Procedure from library primitiv.lib (see primitiv_lib).

Usage:
primitive_extra(i); i ideal

Assume:
The ground field of the basering is k=Q or k=Z/pZ and the ideal i is given by 2 generators f,g with the following properties:
 
   f is the minimal polynomial of a in k[x],
   g is a polynomial in k[x,y] s.th. g(a,y) is the minpoly of b in k(a)[y].
Here, x is the name of the first ring variable, y the name of the second.

Return:
ideal j in k[y] such that
 
   j[1] is the minimal polynomial for a primitive element c of k(a,b) over k,
   j[2] is a polynomial s.th. j[2](c)=a.

Note:
While primitive(i) may fail for finite fields, primitive_extra(i) tries all elements of k(a,b) and, hence, always finds a primitive element.
In order to do this (try all elements), field extensions like Z/pZ(a) are not allowed for the ground field k.
primitive_extra(i) assumes that the second generator, g, is monic as polynomial in (k[x])[y].

Example:
 


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