On the Extensions of a Discrete Valuation in a Number Field

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 $K$ be a number field defined by a monic irreducible polynomial $F(X) \in \mathbb{Z}[X]$, $p$ a fixed rational prime, and $\nu_p$ the discrete valuation associated to $p$. Assume that $\overline{F}(X)$ 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 $\nu_p$. 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.