Elements with prime and small indices in bicyclic biquadratic number fields
Let $K$ be an algebric number field, $\alpha$ an algebraic integer in $K$. The index of $\alpha$ is defined as $I(\alpha)=(\mathbb{Z}_K^+:\mathbb{Z}[\alpha]^+)$. If $I(\alpha)=1$, then $\{1,\alpha,\ldots,\alpha^{n-1}\}$ is an integer basis in $K$ and we call it monogenic.\par Let $m,n$ be distinct square-free integers, consider bicyclic biquadratic fields of type $K=\mathbb{Q}(\sqrt{m},\sqrt{n})$. These fields have an extensive literature which is detailed in the paper.\par The author is interested in the totally complex case of bicyclic biquadratic fields, when $m>0$ and $n<0$. In this case \textit{G. Nyul} [Acta Acad. Paedagog. Agriensis, Sect. Mat. (N. S.) 28, 79--86 (2001; Zbl 0988.11011)] gave necessary and sufficient conditions for the monogenity of these fields.\par In the present paper the author describes all bicyclic biquadratic fields $K$ when $K$ has elements of index $p$ ($p$ being an odd prime), or index at most 10. All element of the given indices are explicitly given.
scientific article; zbMATH DE number 6931204
scientific article