  We consider infinite parametric families of high degree number fields
composed of quadratic fields with pure cubic, pure quartic, pure sextic fields
and with the so called simplest cubic, simplest quartic fields. We explicitly
describe an integral basis of the composite fields. We construct the index
form, describe their factors and prove that the monogenity of the composite
fields imply certain divisibility conditions on the parameters involved. These
conditions usually can not hold, which implies the non-monogenity of the
fields.
  The fields that we consider are higher degree number fields, of degrees 6 up
to 12. The non-monogenity of the number fields is stated very often as a
consequence of the non-existence of the solutions of the index form equation.
Up to our knowlegde it is not at all feasible to solve the index form equation
in these high degree fields, especially not in a parametric form.
  On the other hand our method implies directly the non-monogenity in almost
all cases. We obtain our results in a parametric form, characterizing these
infinite parametric families of composite fields.
