ACTA issues

## Power integral bases in cubic and quartic extensions of real quadratic fields

István Gaál, László Remete

Acta Sci. Math. (Szeged) 85:3-4(2019), 413-429
80/2018

 Abstract. 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. \par We illustrate our method with two detailed examples. DOI: 10.14232/actasm-018-080-z AMS Subject Classification (1991): 11R04; 11D59,11Y50 Keyword(s): monogenic fields, composites of number fields, relative cubic and relative quartic extensions, relative Thue equations Received September 4, 2018 and in final form February 20, 2019. (Registered under 80/2018.)