#### D.4.29.2 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+(y3+y+1)
    ==>    _[3]=x+(y5+y4+y2+1)
    ==>    _[4]=x+(-y5-y4-y3-y2-y-1)
    ==> [2]:
    ==>    1,1,1,1

User manual for Singular version 4.4.1, 2025, generated by texi2html.