Purity and distances between conjugates of elements over henselian valued fields

Abstract

For a henselian valued field $(K,v)$ and a separable-algebraic element $a\in\overline{K}\setminus K$, we consider the set $S_K(a):= \{ v(a-a^\prime) \mid a^\prime\neq a \text{ is a $K$-conjugate of $a$} \}$. The central aim of this paper is to provide a bound for the cardinality of the set $S_K(a)$, and to characterize the elements $a$ for which this set is a singleton. Connections of this set with the notion of \textit{depth} of $a$ has also been explored. We show that $S_K(a)$ is a singleton whenever $K(a)|K$ is a minimal extension. A stronger version of this result is obtained when $a$ has depth one over $K$. We also provide a host of examples illustrating that the bounds obtained are strict. Apart from being of independent interest, another primary motivation for considering this problem comes from the study of ramification ideals. In the depth one case, when $K(a)|K$ is a Galois extension, we obtain intimate connections between the cardinalities of $S_K(a)$ and the number of ramification ideals of the extension $(K(a)|K,v)$. In particular, we show that these cardinalities are same whenever the extension is defectless and non-tame, or whenever $(K,v)$ has rank one. In order to obtain these results, we provide comprehensive descriptions of the ramification ideals of $(K(a)|K,v)$ which extend the known results in this direction.