Hasse's problem for monogenic fields
A classical problem in algebraic number theory is to consider if an algebraic number field admits power integral bases, that is integral bases of type \(1,\alpha,\alpha^2,\ldots,\alpha^{n-1}\), see \textit{I. Gaál} [Diophantine equations and power integral bases, Boston, MA: Birkhäuser (2002; Zbl 1016.11059)]. Such fields are called monogenic. The author gives a survey of their investigations on monogenity of Galois fields, see e.g. \textit{Y. Motoda} and \textit{T. Nakahara} [Arch. Math. 83, No. 4, 309--316 (2004; Zbl 1078.11061)]. Multiquadratic fields \(\mathbb Q(\sqrt{a_1},\ldots,\sqrt{a_r})\) of degree \(2^r\) are especially considered. The case \(r=2\) is known, for \(r=3\) there exists only one specific field that is monogenic, and for \(r\geq 4\) there are no monogenic fields.
scientific article; zbMATH DE number 5554658