Tame fields and distinguished pairs

Abstract

A henselian valued field is called defectless (resp. tame) if each of its finite extensions is defectless (resp. tame). For a henselian field K in [3], a question was posed: “If every simple algebraic extension of K is defectless, then is it true that K is defectless?” An example showed that the answer is “no” in general. In this paper, we show that the analogue of this question for tame fields has an affirmative answer. Indeed it is proved that if every simple algebraic extension of a henselian field K is tame, then it is a tame field. Moreover, for an algebraic element θ over a tame field K, it is known that all those elements appearing in a saturated distinguished chain for θ stay inside of $K(\theta )$. This rises the problem of studying conditions under which the next element to θ in a chain does not necessarily stay inside of $K(\theta )$. Here we will give a sufficient condition for when there is no algebraic element $\alpha \in K(\theta )$ such that $(\theta ,\alpha )$ is a distinguished pair.

Keywords:

• Algebraic field extensions
• distinguished pairs
• henselian valued fields
• tame valued fields

2020 Mathematics Subject Classification:

• 12J10
• 12F05