On the resolution of index form equations in sextic fields with an imaginary quadratic subfield
Let \(K\) be an algebraic number field with ring of integers \(\mathbb{Z}_K\). The index of the primitive integer \(\alpha \in\mathbb{Z}_K\) is defined as the module index \(I(\alpha) = [\mathbb{Z}_K: \mathbb{Z}[a]]\). Solving the equation \(I(\alpha) =\pm 1\) is then equivalent to finding all primitive \(\alpha\in \mathbb{Z}_K\) such that \(1,\alpha, \alpha^2, \dots\) is an integral basis (i.e. a \(\mathbb{Z}\)-basis for \(\mathbb{Z}_K)\). The authors consider the case in which \(K\) is a sextic field containing an imaginary quadratic subfield \(M\). In very many cases an integer \(\theta \in \mathbb{Z}_K\) can be found such that \(\theta\) has relative index 1 over \(M\). Under this extra assumption on the sextic field \(K\), the authors develop an algorithm for solving index form equations \(I(\alpha) =\pm 1\). They show first that such index form equations are reduced to certain cubic Thue equations with integral unknowns in \(M\). These cubic Thue equations are solved, in turn, by a method that makes use of linear forms in logarithms of algebraic numbers, very similar to that of solving ordinary Thue equations, in three stages: (i) Application of a Baker-type theorem (in this case the theorem of Baker and Wüstholz) in order to obtain an upper bound (a very large one) for certain unknown exponents, (ii) Reduction of the upper bound using the LLL basis reduction algorithm and (iii) Enumeration of all solutions below the reduced upper bound; conceptually, a trivial stage, but the most time consuming. Lots of numerical examples are given.
sextic field containing an imaginary quadratic subfield
scientific article; zbMATH DE number 987589