Monogenic biquadratic fields
Let the biquadratic number field \(K\) be given in the form \(K= \mathbb{Q} (\sqrt{dm}, \sqrt{dn})\), where the integers \(d\), \(m\), \(n\) satisfy the following conditions: all are squarefree, pairwise relatively prime and \(dm, dn, nm\neq 1\); \(dm\equiv dn\bmod 4\); \(d>0\), \(m>n\) and \(d<\min \{| m|,| n|\}\) if \(dm\equiv dn\equiv 1\bmod 4\). This representation is unique. Let \(\delta\) be defined such that \(mn\equiv (- 1)^ \delta \bmod 4\). In the present paper several conditions are proved for the monogeneity of \(K\). We cite here only the most general but technically the simplest one. If \(dm\equiv dn\equiv 1\bmod 4\), then \(K\) is non-monogenic. Otherwise \(K\) is monogenic if and only if the conditions \(2^ \delta m+ 2^ \delta n= 4(2^{-\delta} d)\) and the diophantine equation \((u^ 2- v^ 2)^ 2 (2^ \delta m)- (u^ 2+ v^ 2)^ 2 (2^ \delta n)= 4s\), where \(s\in \{1, -1\}\) has a solution in \(u,v\in \mathbb{Z}\) hold. If \(K\) is imaginary then up to finitely many cases \(K\) is non-monogenic and all these exceptional cases are listed in the paper. For the real biquadratic fields the above conditions are in several directions refined and further useful criteria are derived. At the end of the paper extensive tables with parameters of monogenic biquadratic fields can be found. For example all the 303 non-trivial monogenic real fields from the altogether 2393471094 fields with parameters \(1\leq d,m,n\leq 500\) are listed.
scientific article; zbMATH DE number 724676