15.12 Coefficients for Number Fields

Coefficients( z )
Coefficients( F, z )

return the coefficient vector cfs of z with respect to a particular base B, i.e., we have z = cfs * B. If z is the only argument, B is the default base of the default field of z (see DefaultField and Field for Cyclotomics), otherwise F must be a number field containing z, and we have B = F.base.

The default base of a number field is defined as follows:

For the field extension Q_n/Q_m (i.e. both F and F.field are cyclotomic fields), B is the base {cal{B}}_{n,m} described in ZumbroichBase. This is an integral base which is closely related to the internal representation of cyclotomics, thus the coefficients are easy to compute, using only the zumbroichbase fields of F and F.field.

For the field extension L/Q where L is not a cyclotomic field, B is the integral base described in Integral Bases for Number Fields that consists of orbitsums on roots of unity. The computation of coefficients requires the field F.coeffslist.

in future: replace Q by Q_m

In all other cases, B = NormalBaseNumberField( F ). Here, the coefficients of z with respect to B are computed using F.coeffslist and F.coeffsmat.

If F.base is not the default base of F, the coefficients with respect to the default base are multiplied with F.basechangemat. The only possibility where it is allowed to prescribe a base is when the Cyclotomic Field Records).

    gap> F:= NF( [ ER(3), EB(7) ] ) / NF( [ ER(3) ] );
    NF(84,[ 1, 11, 23, 25, 37, 71 ])/NF(12,[ 1, 11 ])
    gap> Coefficients( F, ER(3) ); Coefficients( F, EB(7) );
    [ -E(12)^7+E(12)^11, -E(12)^7+E(12)^11 ]
    [ 11*E(12)^4+7*E(12)^7+11*E(12)^8-7*E(12)^11, 
      -10*E(12)^4-7*E(12)^7-10*E(12)^8+7*E(12)^11 ]
    gap> G:= CF( 8 ); H:= CF( 0, NormalBaseNumberField( G ) );
    CF(8)
    CF( 0,[ 1/4-2*E(8)-E(8)^2-1/2*E(8)^3, 1/4-1/2*E(8)+E(8)^2-2*E(8)^3, 
      1/4+2*E(8)-E(8)^2+1/2*E(8)^3, 1/4+1/2*E(8)+E(8)^2+2*E(8)^3 ])
    gap> Coefficients( G, ER(2) ); Coefficients( H, ER(2) );
    [ 0, 1, 0, -1 ]
    [ -1/3, 1/3, 1/3, -1/3 ]

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