Cubic Relative Extensions

István Gaál
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István Gaál: University of Debrecen, Institute of Mathematics

Chapter Chapter 13 in Diophantine Equations and Power Integral Bases, 2019, pp 207-227 from Springer

Abstract: Abstract In this chapter we consider the question of monogenity in cubic relative extensions. The case of relative cubic extensions considered in Sect. 13.1 are easy in theory, since we just have to solve a cubic relative Thue equation. To make this case more interesting, we tried to go as far as possible and considered (totally real) relative cubic extensions of quintic and even sextic fields. This needs to solve unit equations with 10 and 12 unknown exponents, respectively. This is certainly the limit of applicability of our procedures, especially of Wildanger’s enumeration method. In Sect. 13.2 we consider monogenity in the composite field of two cubic subfields. To settle this problem we have to solve two cubic relative Thue equations over cubic fields and a corresponding third equation. Finally, in Sect. 13.3 we utilize the method of Theorem 1.8 in cubic extensions of real quadratic fields. Here, in addition to solving a relative Thue equation we reduce the problem to solving simple polynomial equations.

Date: 2019
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DOI: 10.1007/978-3-030-23865-0_13

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