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Power integral bases in cubic and quartic extensions of real quadratic fields. (English) Zbl 1449.11104

Summary: Investigations of monogenity and power integral bases were recently extended from the absolute case (over \(\mathbb{Q}\)) to the relative case (over algebraic number fields). Formerly, in the relative case we only succeeded in calculating generators of power integral bases when the ground field is an imaginary quadratic field. This is the first case when we consider monogenity in the more difficult case, in extensions of real quadratic fields. We give efficient algorithms for calculating generators of power integral bases in cubic and quartic extensions of real quadratic fields, more exactly in composites of cubic and quartic fields with real quadratic fields. In case the quartic field is totally complex, we present an especially simple algorithm. We illustrate our method with two detailed examples.

MSC:

11R04 Algebraic numbers; rings of algebraic integers
11D59 Thue-Mahler equations
11Y50 Computer solution of Diophantine equations
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