On the Resolution of Index Form Equations in Sextic Fields with an Imaginary Quadratic Subfield

Abstract

We give an efficient algorithm for the resolution of index form equations, especially for determining power integral bases, in sextic fields with an imaginary quadratic subfield. The method reduces the problem to the resolution of a cubic relative Thue equation over the quadratic subfield. At the end of the paper we give a table containing the generators of all power integral bases in the first 25 fields of this type with smallest discriminant (in absolute value).

On the resolution of relative Thue equations

“…In several cases (see, e.g., [9], [10], [11], [12], [13]), this problem was reduced to relative Thue equations.…”

Power integral bases in prime-power cyclotomic fields

“…Thus, up to equivalence there are at least two generators for O q , namely ζ and ω. When q = 9, Gaál and Pohst [7] prove that up to equivalence there are no additional generators for the ring of integers (the discriminant is −19 683 and the nine generators listed for this discriminant correspond to the six conjugates of ζ and half of the conjugates of ω, since σ a (ω) is equivalent to σ a (ω) = σ 9−a (ω)). It is plausible that there are no additional generators for any prime-power q.…”

Power integral bases in a parametric family of totally real cyclic quintics

“…e.g. [3] for the basic ideas of the algorithm) we solved the unit equation (9) corresponding to the index form equation (5). The solutions allow to express ε/ε and hence also ε, which gives (x 2 , x 3 , x 4 , x 5 ) in view of (8), by taking conjugates and solving the corresponding system of linear equations.…”

“…e.g. [6], [4], [3], [5]). For number fields of higher degree k the problem becomes difficult because of the large degree k(k − 1)/2 of the index form equation and the number of variables k − 1.…”