Introduction

It is a classical problem in algebraic number theory to decide if the ring of integers ℤ K $${\mathbb Z}_K$$ of a number field K is monogenicMonogenic , that is if it admits power integral bases Integral basis power- Power integral basis of type (1, α, …, α n−1). In the 1960s Hasse (Zahlentheorie, Akademie-Verlag, Berlin, 1963, §25.6., p. 438) Hasse, H. asked to give an arithmetic characterization of those number fields which have power integral bases. The first example of a non-monogenic field was given by Dedekind (Abh. König. Ges. der Wissen. zu Göttingen 23:1–23, 1878). Dedekind, R. In this section we recall some basic notions of number fields, and then we give the most important concepts in connection with monogenic fields K having an integral basis (1, α, …, α n−1) that is a power integral basis. We describe this phenomenon both in the absolute case and in the relative case. We also discuss the specialties of the case when K is the composite of two subfields. Our main purpose is to determine generators of power integral bases. As we shall see, this algorithmic problem is satisfactorily solved for lower degree number fields (especially for cubic and quartic fields) and there are efficient methods for certain classes of higher degree fields. Our algorithms enable us in many cases to describe all power integral bases also in infinite parametric families of certain number fields.