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D.8.10.3 roots
Procedure from library zeroset.lib (see zeroset_lib).
- Usage:
- roots(f); where f is a polynomial
- Purpose:
- compute all roots of f in a finite extension of QQ
without multiplicities.
- Return:
- ring, a polynomial ring over an extension field of QQ,
containing a list 'theRoots' and polynomials 'newA' and 'f':
| | - 'theRoots' is the list of roots of the polynomial f (no multiplicities)
- if the ground field is Q(a') and the extension field is Q(a), then
'newA' is the representation of a' in Q(a).
If the basering contains a parameter 'a' and the minpoly remains unchanged
then 'newA' = 'a'.
If the basering does not contain a parameter then 'newA' = 'a' (default).
- 'f' is the polynomial f in Q(a) (a' being substituted by 'newA')
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- Assume:
- ground field to be Q or a simple extension of Q given by a minpoly
Example:
| | LIB "zeroset.lib";
ring R = (0,a), x, lp;
minpoly = a2+1;
poly f = x3 - a;
def R1 = roots(f);
==>
==> // 'roots' created a new ring which contains the list 'theRoots' and
==> // the polynomials 'f' and 'newA'
==> // To access the roots, newA and the new representation of f, type
==> def R = roots(f); setring R; theRoots; newA; f;
==>
setring R1;
minpoly;
==> (a4-a2+1)
newA;
==> (a3)
f;
==> x3+(-a3)
theRoots;
==> [1]:
==> (-a3)
==> [2]:
==> (a)
==> [3]:
==> (a3-a)
map F;
F[1] = theRoots[1];
F(f);
==> 0
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