Computing elements of given index in totally complex cyclic sextic fields (Q1911403)

The author develops a method for solving index form equations in totally complex sextic fields \(K\) which are composites of an imaginary quadratic subfield \(M\) and a totally real cubic subfield \(L\). In a first step the index form equation for \(K\) is reduced to finitely many systems of two norm equations, one from \(K\) over \(M\), the other one from \(L\) over \(\mathbb{Q}\). In a second step the relative norm equations are reduced to cubic Thue inequalities over \(\mathbb{Z}\). The method is then used to demonstrate that in the composite of \(L = \mathbb{Q} (\vartheta_a)\) (for a root \(\vartheta_a\) of \(x^3 - ax^2 - (a+3) x+1=0)\) and \(M = \mathbb{Q} (\omega) \) \((\omega\) generating the maximal order of the imaginary quadratic field \(\mathbb{Q}(\sqrt {-m}))\), the order \(\mathbb{Z} [\vartheta_a, \omega]\) has no power integral basis for \(a \geq 3\) and \(m \geq 19\).