On the Extensions of a Discrete Valuation in a Number Field

2018/02/17 by Abdulaziz Deajim, Deajim, Abdulaziz, Lhoussain El Fadil +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1802.06208

openalex publication_date 2018/02/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K be a number field defined by a monic irreducible polynomial F(X) ∈ ℤ[X], p a fixed rational prime, and ν_p the discrete valuation associated to p. Assume that F(X) factors modulo p into the product of powers of r distinct monic irreducible polynomials. We present in this paper a condition, weaker than the known ones, which guarantees the existence of exactly r valuations of K extending ν_p. We further specify the ramification indices and residue degrees of these extended valuations in such a way that generalizes the known estimates. Some useful remarks and computational examples are also given to highlight some improvements due to our result.

Related

• Polynomial Index in Dedekind Rings
• Integer-valued polynomials on discrete valuation rings of global fields with prescribed lengths of factorizations
• Valuations on K[x] approaching a fixed irreducible polynomial
• Split absolutely irreducible integer-valued polynomials over discrete valuation domains
• Generating sequences of valuations on simple extensions of domains