Key polynomials for simple extensions of valued fields

Let $\iota:K\hookrightarrow L\cong K(x)$ be a simple transcendental extension of valued fields, where $K$ is equipped with valuation $\nu$ rank 1. That is, we assume given 1 and its $\nu'$ to $L$. $(R_\nu,M_\nu,k_\nu)$ denote the ring $\nu$. The purpose this paper present refined version MacLane's theory key polynomials, similar those considered by M. Vaqui\'e, reminiscent related objects studied Abhyankar Moh (approximate roots) T.C. Kuo. Namely, associate $\iota$ countable well ordered set $$ \mathbf{Q}=\{Q_i\}_{i\in\Lambda}\subset K[x]; $Q_i$ are called {\bf polynomials}. Key polynomials which have no immediate predecessor limit $\beta_i=\nu'(Q_i)$. We give an explicit description (which may viewed as generalization Artin--Schreier polynomials). also upper bound on order type polynomials. show that if $\operatorname{char}\ k_\nu=0$ then has at most $\omega$, while in case k_\nu=p>0$ bounded above $\omega\times\omega$, $\omega$ stands for first infinite ordinal.