6.9 CharPol

CharPol( z )
CharPol( F, z )

In the first form CharPol returns the coefficients of the characteristic polynomial of the element z in its default field over its prime field (see DefaultField). In the second form CharPol returns the coefficients of the characteristic polynomial of the element z in the field F over the subfield F.field. The characteristic polynomial is returned as a list of coefficients, the i-th entry is the coefficient of x^{i-1}.

The characteristic polynomial of an element z in a field F over a subfield S is the frac{[F:S]}{{rm deg } mu}-th power of mu, where mu denotes the minimal polynomial of z in F over S. It is fixed under the Galois group of the normal closure of F. Thus all the coefficients of the characteristic polynomial lie in S. The constant term is (-1)^{F.degree/S.degree}=(-1)^{[F:S]} times the norm of z (see Norm), and the coefficient of the second highest degree term is the negative of the trace of z (see Trace). The roots (including their multiplicities) in F of the characteristic polynomial of z in F are the conjugates (see Conjugates) of z in F.

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

The default function FieldOps.CharPol multiplies the linear factors x - c with c ranging over the conjugates of z in F (see Conjugates). For nonabelian extensions, it is overlayed by a function, which computes the appropriate power of the minimal polynomial.

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