The algebraic numbers definable in various exponential fields

TL;DR

This paper investigates the definability of algebraic numbers within exponential fields, establishing that in certain fields all real abelian algebraic numbers are definable, while in Zilber fields only these numbers are definable.

Contribution

It proves that in E-fields with cyclic kernel, all real abelian algebraic numbers are definable, and characterizes the algebraic numbers definable in Zilber fields.

Findings

All real abelian algebraic numbers are pointwise definable in E-fields with cyclic kernel.

In Zilber fields, only real abelian algebraic numbers are pointwise definable.

The results clarify the structure of definable algebraic numbers in exponential fields.

Abstract

We prove the following theorems: Theorem 1: For any E-field with cyclic kernel, in particular $\mathbb C$ or the Zilber fields, all real abelian algebraic numbers are pointwise definable. Theorem 2: For the Zilber fields, the only pointwise definable algebraic numbers are the real abelian numbers.