On biquadratic fields that admit unit power integral basis

Abstract

We consider biquadratic number fields whose maximal orders have power integral bases consisting of units. We prove an effective and efficient criteria to decide whether the maximal order of a biquadratic field has a unit power integral basis or not. In particular we can determine all trivial biquadratic fields whose maximal orders have a unit power integral basis.

On a family of biquadratic fields that do not admit a unit power integral basis

“…The proof of Theorem 1 follows closely the strategy due to Pethő and Ziegler in [11]. However, we have to sharpen considerably the bounds found in [11] to obtain Theorem 1. Let us give a short overview of the strategy.…”

“…In the final section, we show that the potential two solutions to (1) cannot come both from θ and θ −1 , so we are left to consider only "small" solutions. However, by the vast improvements in this paper, the bounds are considerably smaller than those provided in [11], and we succeed in proving Theorem 1.…”

“…ξ . This idea has been applied successfully in the case of maximal orders of biquadratic number fields by Pethő and Ziegler [11]. In the case of biquadratic number fields, we only have a criterion which can be hard to apply.…”

Additive unit representations in rings over global fields -- A survey

“…Pethő and Ziegler investigated a modified version of Problem A, where one asks whether a ring of integers has a power basis consisting of units [39,29]. For example, Ziegler proved the following: For rings R with u(R) = ω, Ashrafi [1] investigated the stronger property that every element of R can be written as a sum of k units for all sufficiently large integers k. Ashrafi proved that this is the case if and only if R does not have Z/2Z as a factor, and applied this result to rings of integers of quadratic and complex cubic number fields.…”

Representing algebraic integers as linear combinations of units

“…For results in this direction we refer to a paper of Pethő and Ziegler [17]. Showing that (up to certain precisely described exceptions) every number field admits a basis consisting of units with small conjugates, we prove that allowing a small, completely explicit set of (rational) coefficients every integer of K can be expressed as a linear combination of units.…”