On the distances of an element to its conjugates

Abstract: For a valued field $(K,v)$, with a fixed extension of $v$ to the algebraic closure $\overline K$ of $K$, and an element $\theta\in\overline K$, we are interested in the possible values of $\theta-\theta'$ where $\theta'$ runs through all the $K$-conjugates of $\theta$. The study of these values is a classic problem in number theory and ramification theory. However, the classic results focus on tame, and in particular defectless, extensions. In this paper we focus on the study of defect extensions. We want to compare the number of such values to invariants of $\theta$. The main invariant we have in mind is the depth of $\theta$. We present various examples that show that, in the defect case, none of the equivalent of the classic results are true. We also discuss the relation between the number of such values and the number of ramification ideals of the extension $(K(\theta)/K,v)$. In order to do so, we present some results about ramification ideals that have interest on their own.