On The Geometric Determination of Extensions of Non-Archimedean Absolute Values

Let | | be a discrete non-archimedean absolute value of a field with valuation ring O, maximal ideal and residue field F = . Let be a simple finite extension of generated by a root of a monic irreducible polynomial ∈ ]. Assume that in F[ ] for some monic polynomial ∈ ] whose reduction modulo is irreducible, the -Newton polygon has a single side of negative slope λ, and the residual polynomial )( ) has no multiple factors in F ]. In this paper, we describe all absolute values of 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.