A Point-Free Look at Ostrowski's Theorem and Absolute Values

TL;DR

This paper provides a constructive, point-free topological analysis of absolute values on integers, extending Ostrowski's Theorem to include prime ideals and seminorms, with implications for non-Archimedean spectra.

Contribution

It introduces a point-free, topos-theoretic framework for absolute values on integers, generalizing Ostrowski's Theorem and connecting to Berkovich and adic spectra.

Findings

Homeomorphism between absolute values and prime ideals with upper reals

Extension of Ostrowski's Theorem to include multiplicative seminorms

Constructive, geometric approach using topos theory

Abstract

This paper investigates the absolute values on $\mathbb{Z}$ valued in the upper reals (i.e. reals for which only a right Dedekind section is given). These necessarily include multiplicative seminorms corresponding to the finite prime fields $\mathbb{F}_p$. As an Ostrowski-type Theorem, the space of such absolute values is homeomorphic to a space of prime ideals (with co-Zariski topology) suitably paired with upper reals in the range $[-\infty, 1]$, and from this is recovered the standard Ostrowski's Theorem for absolute values on $\mathbb{Q}$. Our approach is fully constructive, using, in the topos-theoretic sense, geometric reasoning with point-free spaces, and that calls for a careful distinction between Dedekinds vs. upper reals. This forces attention on topological subtleties that are obscured in the classical treatment. In particular, the admission of multiplicative seminorms points to connections with Berkovich and adic spectra. The results are also intended to contribute to characterising a (point-free) space of places of $\mathbb{Q}$.