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

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 $\mathbb{Q}$.