D.7.5.5 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 the ground field without multiplicities.

Return:

ring, a polynomial ring over an extension field of the ground field, containing a list ’roots’ and polynomials ’newA’ and ’f’:

- ’roots’ 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’)

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 'roots' and
→ // the polynomials 'f' and 'newA'
→ // To access the roots, newA and the new representation of f, type
→    def R = Roots(f); setring R; roots; newA; f;
→ 
setring R1;
minpoly;
→ (a4-a2+1)
newA;
→ (a3)
f;
→ x3+(-a3)
roots;
→ [1]:
→    (-a3)
→ [2]:
→    (a3-a)
→ [3]:
→    (a)
map F;
F[1] = roots[1];
F(f);
→ 0