Differential exponential topological fields

TL;DR

This paper develops an axiomatic framework for exponential fields with derivations, applying it to real and p-adic number fields to explore their algebraic and topological properties.

Contribution

It introduces a new class of existentially closed exponential fields with derivations and applies this theory to real and p-adic number systems.

Findings

Axiomatization of exponential fields with derivations

Application to real exponential functions

Application to p-adic exponential functions

Abstract

We axiomatize a class of existentially closed exponential fields equipped with an $E$-derivation. We apply our results to the field of real numbers endowed with $exp(x)$ the classical exponential function defined by its power series expansion and to the field of p-adic numbers endowed with the function $exp(px)$ defined on the $p$-adic integers where $p$ is a prime number strictly bigger than $2$ (or with $exp(4x)$ when $p=2$).