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

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-Vaquié 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.