
D.4.23.2 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+y21;
primitive_extra(i);
==> _[1]=y6y5+y4y3y1
==> _[2]=y5+y4+y2+y+1
ring extension=(3,y),x,dp;
minpoly=y6y5+y4y3y1;
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+(y5y4y2y1)
==> [2]:
==> 1,1,1
factorize(x3+x21);
==> [1]:
==> _[1]=1
==> _[2]=x+(y5y4y3y2y1)
==> _[3]=x+(y5+y4+y2+1)
==> _[4]=x+(y3+y+1)
==> [2]:
==> 1,1,1,1

