Index form equations in quintic fields

Index form equations in sextic fields: a hard computation

“…Just as in [6] we can apply Baker's type estimates, that is, we can calculate positive constants c 1 , c 2 and C 0 (this latter constant is a huge one given for example by the estimates of A. Baker and G. Wüstholz [1]) such that for certain distinct indices i, j, k,…”

“…As in [6] this covers the most difficult (hence most interesting) cases. In our example the field has Galois group S 6 .…”

“…Our enumeration process is divided into several steps similar to those in [6]. Instead of Lemma 2 in [6] we shall use the following consequence of our Lemma 2 above: Lemma 3.…”

“…In the last decade considerable effort was made to develop efficient algorithms for finding explicitly the solutions, which was successful for lower degree fields (n ≤ 5) and partially successful for higher degree fields; see [4]. I. Gaál and K. Győry [6] gave an algorithm for solving index form equations in arbitrary quintic fields. Despite the very short CPU times we had for cubic and quartic fields, in the quintic case (in the most interesting case of totally real fields with Galois group S 5 ) about 8 hours of CPU time were necessary (using a 1 GHz PC under Linux).…”

“…Despite the very short CPU times we had for cubic and quartic fields, in the quintic case (in the most interesting case of totally real fields with Galois group S 5 ) about 8 hours of CPU time were necessary (using a 1 GHz PC under Linux). In [6] the ideas of K. Győry [8], [9] were used in reducing the index form equation to appropriate unit equations in two variables and applying Baker's estimates. It was observed that if K = Q(ξ) is a totally real quintic field with a doubly transitive Galois group (which is satisfied in the most difficult cases), then the values of the linear factors of the index form are contained in fields of type…”