Monogenity in totally real extensions of imaginary quadratic fields with an application to simplest quartic fields

We describe an efficient algorithm to calculate generators of power integral bases in composites of totally real fields with imaginary quadratic fields. We show that the calculation can be reduced to solving index form equations in the original totally real fields. We illustrate our method by investigating monogenity in the infinite parametric family of imaginary quadratic extensions of the simplest quartic fields.