A note on saturated distinguished chains of length one

Abstract

Let [Formula: see text] be a Henselian valued field. In this paper, we investigate algebraic elements [Formula: see text] having saturated distinguished chains of length one over [Formula: see text]. We provide some characterizations for them. In particular, we characterize all irreducible polynomials over [Formula: see text] whose roots have saturated distinguished chains of length one. Moreover, using the properties of such elements we give various necessary, sufficient, or both conditions for the metric invariants of algebraic elements to be equal.