Monogenity of Biquadratic Fields Related to Dedekind-Hasse’s Problem

Abstract

The aim of this paper is to determine the monogenity of imaginary, and real biquadratic fields K over the field Q of rational numbers and the relative monogenity of K over its quadratic subfield k. To characterize such phenomena it is necessary to determine an integral basis of the field K and to evaluate the relative norm of the different d(ξ) with respect to K/k of an integer ξ in K. Here d(ξ) is defined by Q ρ∈G\{ι} (ξ − ξ ρ ), where ξ − ξ ρ denotes the partial different of an integer ξ in K, and G and ι denote the Galois group of K/Q and the identity embedding of K, respectively. For the succinct proof of non-monogenity, we consider a single linear Diophantine equation consisted of the partial differents with unit coefficients.