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Sextic Fields

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Diophantine Equations and Power Integral Bases

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Abstract

In Sect. 11.1 we shall summarize how the analogue of the method of Sect. 10.1 can be applied for sextic fields. Further we present much more efficient methods to calculate generators of power integral bases in case the sextic field admits some additional property, making the index form equation easier. We have efficient algorithms for sextic fields having quadratic or cubic subfields (see Sects. 11.2 and 11.3). Investigating the structure of the index form in sextic fields with a quadratic subfield we shall point out the important role of various types of Thue equations (see Gaál, Application of Thue equations to computing power integral bases in algebraic number fields. In: Proceedings of the Conference on ANTS II. Lecture notes in computer science, vol 1122. Springer, Berlin, pp 151–155, 1996). In Sect. 11.4 we shall consider sextic fields that are composites of a quadratic and a cubic field. We show some interesting applications of the results of Sect. 7.5.2 on composite fields. We close the chapter by investigating power integral bases in the infinite parametric family of simplest sextic fields (Sect. 11.5).

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Authors and Affiliations

  1. Institute of Mathematics, University of Debrecen, Debrecen, Debrecen, Hungary

    István Gaál

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  1. István Gaál
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Gaál, I. (2019). Sextic Fields. In: Diophantine Equations and Power Integral Bases. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-23865-0_11

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