GAP Manual: 15.11. NormalBaseNumberField

NormalBaseNumberField( F )
NormalBaseNumberField( F, x )

returns a list of cyclotomics which form a normal base of the number field F (see Number Field Records), i.e. a vector space base of the field F over its subfield F.field which is closed under the action of the Galois group F.galoisGroup of the field extension.

The normal base is computed as described in Art68: Let Phi denote the polynomial of a field extension L/L^(prime), Phi^(prime) its derivative and alpha one of its roots; then for all except finitely many elements zinL^(prime), the conjugates of frac(Phi(z))((z-alpha)cdotPhi^(prime)(alpha)) form a normal base of L/L^(prime).

When NormalBaseNumberField( F ) is called, z is chosen as integer, starting with 1, NormalBaseNumberField( F, x ) starts with z=x, increasing by one, until a normal base is found. 

    gap> NormalBaseNumberField( CF( 5 ) );
    [ -E(5), -E(5)^2, -E(5)^3, -E(5)^4 ]
    gap> NormalBaseNumberField( CF( 8 ) );
    [ 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 ]
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