When is scalar multiplication decidable?

TL;DR

This paper characterizes when the theory of real ordered vector spaces expanded by integers is decidable, showing it depends precisely on whether the subfield is a real quadratic field.

Contribution

It establishes a complete criterion for decidability of the theory based on the nature of the subfield $K$, specifically identifying real quadratic fields as the key case.

Findings

Decidability holds if and only if $K$ is a real quadratic field.

The theory is undecidable for other subfields of $ eal$.

Provides a clear classification linking field properties to logical decidability.

Abstract

Let $K$ be a subfield of $\mathbb{R}$. The theory of $\mathbb{R}$ viewed as an ordered $K$-vector space and expanded by a predicate for $\mathbb{Z}$ is decidable if and only if $K$ is a real quadratic field.