Cuts of a totally ordered field of rational functions over an Archimedean field

Abstract

We study the classification of Dedekind cuts in the field K(α), where K ⊆ ℝ and α is a positive infinitesimal. The cuts are analyzed according to three criteria: fundamentality, symmetry, and algebraicity. We prove that every non-principal cut in K(α) that is both non-fundamental and asymmetric must be algebraic. For such cuts, we construct a sign-changing polynomial whose root realizes the cut. Furthermore, we investigate the properties of these polynomials and their dependence on the structure of the base field K. The results contribute to the broader understanding of algebraic and order-theoretic properties in non-Archimedean extensions of real fields, particularly in the context of model theory and real algebraic geometry. The classification developed here provides a constructive approach to identifying algebraic cuts and offers insights into the interaction between infinitesimal elements and the topological structure of real closed fields.