On unit power integral bases of \(\mathbb Z[\root 4 \of {m}]\) (Q663103)

A classical problem in computational algebraic number theory is to find an integral basis to a given number field. There are many algorithms to find such integral bases for a given number field. In many cases it is of interest to know whether a power integral basis exists and if so to have an algorithm to compute it. This kind of problem leads to index form equations which were studied by several authors. Another problem of algebraic number theory is to investigate the additive properties of units of number fields, that is, can every algebraic integer of a given order of a number field be written as the sum of units. Combining these two topics the author asks the following: ``For which number fields does there exist a power integral basis consisting of units?'' The aim of this paper is to solve this problem for orders of the type \(\mathbb Z[\root 4 \of m]\). Let \(m \neq 0\) be an integer which is not a perfect square and consider number fields of the form \(\mathbb Q(\root 4 \of m)\). The author characterizes all orders of the form \(\mathbb Z[\root 4 \of m]\) which admit a unit power integral basis, i.e., there exists a unit \(\varepsilon\) such that \(1,\varepsilon,\varepsilon^2\) and \(\varepsilon^3\) is an integral basis of \(\mathbb Z[\root 4 \of m]\).