A Dedekind's Criterion over Valued Fields

2019/08/18 by Lhoussain El Fadil, Fadil, Lhoussain El, Mhammed Boulagouaz +3
Computer Science · Mathematics · #12J10 #13A18 #13B22 #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1908.06365

openalex publication_date 2019/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (K,ν) be an arbitrary-rank valued field, R_ν its valuation ring, K(α)/K a separable finite field extension generated over K by a root of a monic irreducible polynomial f∈ R_ν[X]. We give necessary and sufficient conditions for R_ν[α] to be integrally closed. We further characterize the integral closedness of R_ν[α] based on information about the valuations on K(α) extending ν. Our results enhance and generalize some existing results in the relevant literature. Some applications and examples are also given.

Related

• On a Theorem of Dedekind
• On the Integral Closedness of R[α]
• Polynomial Index in Dedekind Rings
• On common extensions of valued fields
• Generating sequences of valuations on simple extensions of domains