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

Let | | be a discrete non-archimedean absolute value of a field K with valuation ring 𝒪, maximal ideal 𝓜 and residue field 𝔽 = 𝒪/𝓜. Let L be a simple finite extension of K generated by a root α of a monic irreducible polynomial F ∈ O[x]. Assume that F¯=ϕ¯l$\overline F = \overline \varphi ^l$ in 𝔽[x] for some monic polynomial φ ∈ O[x] whose reduction modulo 𝓜 is irreducible, the φ-Newton polygon Nφ¯(F)$N\overline \phi \left( F \right)$ has a single side of negative slope λ, and the residual polynomial R_λ(F )(y) has no multiple factors in 𝔽_φ[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.