On the distances of an element to its conjugates

TL;DR

This paper investigates the values of differences between an element and its conjugates in valued field extensions, especially focusing on defect extensions where classic results do not hold, and explores their relation to ramification invariants.

Contribution

It provides new insights into defect extensions in valued fields, contrasting them with tame cases, and introduces examples and results relating conjugate differences to ramification invariants.

Findings

Classic results do not hold in defect extensions.

The number of conjugate differences relates to ramification ideals.

Examples illustrate the distinct behavior in defect cases.

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.