Integral Bases and Monogenity of Composite Fields

ABSTRACT

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 cannot 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. As per our knowledge, 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.

KEYWORDS:

• composite fields
• integral basis
• monogenity
• power integral basis
• pure fields
• simplest cubic fields
• simplest quartic fields

2010 AMS Subject Classification:

• Primary 11R04
• Secondary 11Y50