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D.4.20.1 primitive
Procedure from library primitiv.lib (see primitiv_lib).
- Usage:
- primitive(i); i ideal
- Assume:
- i is given by generators m[1],...,m[n] such that for j=1,...,n
- m[j] is a polynomial in k[x(1),...,x(j)]
- m[j](a[1],...,a[j-1],x(j)) is the minimal polynomial for a[j] over
k(a[1],...,a[j-1])
(k the ground field of the current basering and x(1),...,x(n)
the ring variables).
- Return:
- ideal j in k[x(n)] with
- j[1] a minimal polynomial for a primitive element b of
k(a[1],...,a[n]) over k,
- j[2],...,j[n+1] polynomials in k[x(n)] such that j[i+1](b)=a[i]
for i=1,...,n.
- Note:
- the number of variables in the basering has to be exactly n,
the number of given generators (i.e., minimal polynomials).
If the ground field k has only a few elements it may happen that no
linear combination of a[1],...,a[n] is a primitive element. In this
case primitive(i) returns the zero ideal, and one should use
primitive_extra(i) instead.
Example:
See also:
primitive_extra.
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