Power integral bases in algebraic number fields whose Galois groups are 2-elementary abelian
Let \(K\) be a number field and \(O_K\) its ring of integers. We say that the extension \(K/\mathbb {Q}\) is monogenic if \(O_K\) has a power integral basis, that is, there exists \(\theta \in O_K\) such that \(O_K=\mathbb {Z}[\theta]\). Let \(K/\mathbb {Q}\) be a Galois extension whose Galois group is \(2\)-elementary abelian of order \(2^r\). When \(r=2\), in [\textit{M. N. Gras}, and \textit{F. Tanoé}, Manuscr. Math. 86, No.~1, 63--79 (1995; Zbl 0816.11058)] a necessary and sufficient condition for \(K/\mathbb {Q}\) to be monogenic is given. In the paper under review, the authors suppose \(r\geq 3\). If \(r\geq 4\), an immediate consequence of a result in [\textit{S. I. A. Shah, T. Nakahara}, Nagoya Math. J. 168, 85--92 (2002; Zbl 1036.11052)] is that \(K/\mathbb {Q}\) is not monogenic. Now suppose \(r=3\). The main result of the paper is the following: under some technical hypotheses, \(K/\mathbb {Q}\) is monogenic if and only if \(K=\mathbb {Q}(\sqrt {-1}, \sqrt {-2}, \sqrt {-3})\).
scientific article; zbMATH DE number 2130538