Ideal constructions and irrationality measures of roots of algebraic numbers

This paper addresses the problem of determining the best results one can expect using the Thue-Siegel method as developed by Bombieri in his equivariant approach to effective irrationality measures to roots of high order of algebraic numbers, in the non-archimedean setting. As an application, we show that this method, under a non-vanishing assumption for the auxiliary polynomial which replaces the appeal to Dyson's Lemma type arguments and together with a version of Siegel's Lemma due to Struppeck and Vaaler, yields a result comparable to the best results obtained to date by transcendence methods.