On the first-order theories of quaternions and octonions

Abstract

Let $L$ be the language of rings. We provide an axiomatization of the $L$-theories of quaternions and octonions and characterize their models: they coincide, up to isomorphism, with quaternion and octonion algebras over a real closed field, respectively. We bi-interpret these theories in terms of real closed fields and we prove they are complete, model complete and they do not have quantifier elimination. Then, we focus on the class of ordered polynomials. Over $\mathbb{H}$ and $\mathbb{O}$ these polynomials are of special interest in hypercomplex analysis since they are slice regular. We deduce some fundamental properties of their zero loci from model completeness and we introduce the notions of algebraic sets and Zariski topology. Finally, we prove the failure of quantifier elimination for the fragment of ordered formulas and we completely characterize the family of algebraic sets.