Q:=RationalField();
Pol<X>:=PolynomialRing(Q);
K<v>:=NumberField(X^2-2);  // v = square root of 2.
L<w>:=NumberField(PolynomialRing(K)!X^3-2); // w = cubic root of 2.
// By writing "PolynomialRing(K)!X^3-2" we ask magma to view X^3-2 as a polynomial over the field K.
// This means that L is considered as an extension of K.
"Basis K/Q =",Basis(K);   // Basis of K/Q
"[K:Q] =",Degree(K);
"Basis L/M =",Basis(L);   // Basis of L/K
"[L:M] =",Degree(L);
//
// Now we ask magma to compute the basis of L/Q. The characterization "absolute" means "over Q".
"Basis L/Q =",AbsoluteBasis(L);
//
// Let us take an element "mixing" v and w, for example:
z1:=(v+w)/(1+v+w^2);
"z1 =",z1;
// Observe that z is expressed as a linear combination of the elements of the "AbsoluteBasis(L)" with coefficients in Q.

z2:=((2+3*v)+(1/5-3/4*v)*w+(1/100-12*v)*w^2)/((1/2-2/3*v)+(1-31*v)*w+(2/5-7/8*v)*w^2);
"z2 =",z2;