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D.4.30.2 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.
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- 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
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