D.4.8.2 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:

LIB "primitiv.lib";
ring exring=3,(x,y),dp;
ideal i=x2+1,y3+y2-1;
primitive_extra(i);
→ _[1]=y6-y5+y4-y3-y-1
→ _[2]=y5+y4+y2+y+1
ring extension=(3,y),x,dp;
minpoly=y6-y5+y4-y3-y-1;
number a=y5+y4+y2+y+1;
a^2;
→ -1
factorize(x2+1);
→ [1]:
→    _[1]=1
→    _[2]=x+(-y5-y4-y2-y-1)
→    _[3]=x+(y5+y4+y2+y+1)
→ [2]:
→    1,1,1
factorize(x3+x2-1);
→ [1]:
→    _[1]=1
→    _[2]=x+(-y5-y4-y3-y2-y-1)
→    _[3]=x+(y5+y4+y2+1)
→    _[4]=x+(y3+y+1)
→ [2]:
→    1,1,1,1