Tame fields, Graded Rings and Finite Complete Sequences of Key Polynomials

TL;DR

This paper characterizes tame valued fields using finite complete sequences of key polynomials, linking field properties to graded ring structures and extending valuation theory.

Contribution

It introduces a criterion for tame fields via key polynomials and connects valuation properties to graded ring Frobenius endomorphism.

Findings

Valued field $(K,v)$ is tame iff $vK$ is $p$-divisible, $Kv$ perfect, and extensions admit finite key polynomial sequences.

Characterization of defectless and henselian fields using Mac Lane-Vaquié chains.

Analysis of properties like simply defectless, algebraically maximal, inertial, and ramified extensions.

Abstract

In this paper, we present a criterion for $(K,v)$ to be henselian and defectless in terms of finite complete sequences of key polynomials. For this, we use the theory of Mac Lane-Vaqui\'e chains and abstract key polynomials. We then prove that a valued field $(K,v)$ is tame if and only if $vK$ is $p$-divisible, $Kv$ is perfect and every simple algebraic extension of $K$ admits a finite complete sequence of key polynomials. The properties $vK$ $p$-divisible and $Kv$ perfect are described by the Frobenius endomorphism on the associated graded ring. We also make considerations on simply defectless and algebraically maximal valued fields and purely inertial and purely ramified extensions.