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Hasse’s problem for monogenic fields. (English) Zbl 1187.11038

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 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. Y. Motoda and 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.

MSC:

11R04 Algebraic numbers; rings of algebraic integers
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References:

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