# 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 z in L^{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 ]```

GAP 3.4.4
April 1997