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

Japhet Odjoumani; Alain Togbé; Volker Ziegler

2019/94/1-2 (1) — DOI: 10.5486/PMD.2019.8103 — pp. 1-19

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

Authors: Japhet Odjoumani, Alain Togbé and Volker Ziegler

Abstract:

In this paper, we consider the following family of biquadratic fields: $\mathbb{K}=\mathbb{Q}(\sqrt{18n^2+17n+4},\sqrt{2n^2+n})$, and show that provided that $18n^2+17n+4$ and $2n^2+n$ are both square-free, $\mathbb{K}$ does not admit a power integral basis consisting of units.

Keywords: unit sum number problem, power integral basis, system of Pell equations

Mathematics Subject Classification: 11R16, 11D57, 11R33