Quartic Relative Extensions

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

Chapter Chapter 14 in Diophantine Equations and Power Integral Bases, 2019, pp 229-264 from Springer

Abstract: Abstract In this chapter we consider quartic relative extensions, relative power integral bases in these extensions, and also the absolute power integral bases in the extension field by using the relative power integral bases. In Sect. 14.1 we describe a relative analogue of the method of Sect. 9.1 to calculate relative power integral bases in relative quartic extensions. Applying this method in Sect. 14.2 we consider power integral bases in octic fields with quadratic subfields by using the relative power integral bases. We consider the more interesting case of real quadratic subfields. In Sect. 14.3 we show how much easier is the problem if we consider composites of quartic fields with real quadratic fields. In case the quadratic field is totally complex, in Sect. 14.3.1 we give a relative analogue of the method of Sect. 9.4 using quadratic forms. In the rest of the chapter we consider relative quartic extensions of imaginary quadratic fields. Then the calculation is easier and we obtain results on a wider class of number fields. In Sect. 14.4 we consider relative and absolute power integral bases of pure quartic extensions of imaginary quadratic fields. In Sect. 14.5 we consider monogenity in there infinite parametric families of octic fields with imaginary quadratic subfields using various methods which gives a good overview about the possible tools.

Date: 2019
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DOI: 10.1007/978-3-030-23865-0_14

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