On the Geometric Determination of Extensions of non-Archimedean Absolute Values

Abstract

Let $|\;|$ be a  discrete  non-archimedean absolute value of a field $K$ with valuation ring $\Oo$, maximal ideal $\mathcal{M}$ and residue field $\F=\Oo/\mathcal{M}$. Let $L$ be a simple finite  extension of $K$ generated by  a root  $\alpha$ of a monic irreducible polynomial $F\in \Oo[x]$.   Assume that $\ol{F}=\ol{\ph}^{l}$ in $\F[x]$  for some monic  polynomial $\phi\in\Oo[x]$ whose reduction modulo $\mathcal{M}$ is  irreducible, the $\ph$-Newton polygon $\npp{F}$ has a single side of negative slope $\la$, and the residual polynomial $R_\la(F)(y)$ has no multiple factors in $\F_\phi[y]$. In this paper, we describe all absolute values of $L$ extending $|\ |$. The problem is classical but our approach uses new ideas. Some useful remarks and computational examples are given to highlight some improvements due to our results.