6.8 MinPol

MinPol( z )
MinPol( F, z )

In the first form MinPol returns the coefficients of the minimal polynomial of the element z in its default field over its prime field (see DefaultField). In the second form MinPol returns the coefficients of the minimal polynomial of the element z in the field F over the subfield F.field.

Let F/S be a field extension and L a minimal normal extension of S, containing F. The minimal polynomial of z in F over S is the squarefree polynomial whose roots are precisely the conjugates of z in L (see Conjugates). Because the set of conjugates is fixed under the Galois group of L over S (see GaloisGroup), so is the polynomial. Thus all the coefficients of the minimal polynomial lie in S.

    gap> MinPol( Z(2^6) );
    [ Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0 ]
    gap> MinPol( GF(2^12), Z(2^6) );
    [ Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0, Z(2)^0, 0*Z(2), Z(2)^0 ]
    gap> MinPol( GF(2^12)/GF(2^2), Z(2^6) );
    [ Z(2^2), Z(2)^0, Z(2)^0, Z(2)^0 ] 

The default function FieldOps.MinPol, which works only for extensions with abelian Galois group, multiplies the linear factors x - c with c ranging over the set of conjugates of z in F (see Conjugates). For generic algebraic extensions, it is overlayed by solving a system of linear equations, given by the coefficients of powers of z in respect to a given base.

Previous Up Top Next
Index