On the Extensions of a Discrete Valuation in a Number Field
Abdulaziz Deajim, Lhoussain El Fadil

TL;DR
This paper provides a new, weaker condition for determining the number of valuations extending a discrete valuation in a number field, along with detailed ramification and residue degree information, supported by examples.
Contribution
It introduces a less restrictive criterion for the existence and properties of valuations extending a discrete valuation in a number field.
Findings
Established a weaker condition for valuation extension existence.
Generalized estimates for ramification indices and residue degrees.
Provided computational examples demonstrating improvements.
Abstract
Let be a number field defined by a monic irreducible polynomial , a fixed rational prime, and the discrete valuation associated to . Assume that factors modulo into the product of powers of distinct monic irreducible polynomials. We present in this paper a condition, weaker than the known ones, which guarantees the existence of exactly valuations of extending . 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.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Analytic Number Theory Research · Advanced Differential Equations and Dynamical Systems
