#### 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')

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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