Totally real Thue inequalities over imaginary quadratic fields

Let $F(x,y)$ be an irreducible binary form of degree $\geq 3$ with integer coefficients and with real roots. Let $M$ be an imaginary quadratic field, with ring of integers $Z_M$. Let $K>0$. We describe an efficient method how to reduce the resolution of the relative Thue inequalities $$|F(x,y)|\leq K \;\; (x,y\in Z_M)$$ to the resolution of absolute Thue inequalities of type $$|F(x,y)|\leq k \;\; (x,y\in Z).$$ We illustrate our method with an explicit example

Monogenity in totally complex sextic fields, revisited

In addition to rather complicated general methods it is interesting and valuable to develop fast efficient methods for calculating generators of power integral bases in special types of number fields. We consider sextic fields containing a real cubic and a complex quadratic fields. We develop a very simple and very efficient method to calculate generators of power integral bases in this type of fields. Our method can be applied to infinite families of number fields, as well. We substantially improve the former methods. Our algorithm is illustrated with detailed examples, involving infinite parametric families