Ordered exponential field
# Ordered exponential field

In mathematics, an ordered exponential field is an ordered field together with a function which generalises the idea of exponential functions on the ordered field of real numbers.

## Definition

An exponential E{\textstyle E} on an ordered field K{\textstyle K} is a strictly increasing isomorphism of the additive group of K{\textstyle K} onto the multiplicative group of positive elements of K{\textstyle K}. The ordered field $K\,$ together with the additional function $E\,$ is called an ordered exponential field.

## Examples

* The canonical example for an ordered exponential field is the ordered field of real numbers R with any function of the form $a^x{\textstyle$ a^{x}} where a{\textstyle a} is a real number greater than 1. One such function is the usual exponential function, that is E(x) = $e^x.$ The ordered field R equipped with this function gives the ordered real exponential field, denoted by $R_{exp}.$ It was proved in the 1990s that $R_{exp}$ is model complete, a result known as Wilkie's theorem. This result, when combined with Khovanskiĭ's theorem on pfaffian functions, proves that $R_{exp}$ is also o-minimal. Alfred Tarski posed the question of the decidability of $R_{exp}$ and hence it is now known as Tarski's exponential function problem. It is known that if the real version of Schanuel's conjecture is true then $R_{exp}$ is decidable.
* The ordered field of surreal numbers No{\textstyle \mathbf {No} } admits an exponential which extends the exponential function exp on R. Since No{\textstyle \mathbf {No} } does not have the Archimedean property, this is an example of a non-Archimedean ordered exponential field.
* The ordered field of logarithmic-exponential transseries $T^{LE}{\textstyle$ \mathbb {T} ^{LE}} is constructed specifically in a way such that it admits a canonical exponential.

## Formally exponential fields

A formally exponential field, also called an exponentially closed field, is an ordered field that can be equipped with an exponential E{\textstyle E}. For any formally exponential field K{\textstyle K}, one can choose an exponential E{\textstyle E} on K{\textstyle K} such that 1+1/n<E(1)<n{\textstyle 1+1/n<E(1)<n} for some natural number n{\textstyle n}.

## Properties

* Every ordered exponential field K{\textstyle K} is root-closed, i.e., every positive element of $K\,$ has an n{\textstyle n}-th root for all positive integer n{\textstyle n} (or in other words the multiplicative group of positive elements of $K\,$ is divisible). This is so because E( 1/n $E^{−1}(a))^n=E(E^{−1}(a))=a{\textstyle$ E\left({\frac {1}{n}}E^{-1}(a)\right)^{n}=E(E^{-1}(a))=a} for all a>0{\textstyle a>0}.
* Consequently, every ordered exponential field is a Euclidean field.
* Consequently, every ordered exponential field is an ordered Pythagorean field.
* Not every real-closed field is a formally exponential field, e.g., the field of real algebraic numbers does not admit an exponential. This is so because an exponential E{\textstyle E} has to be of the form $E(x)=a^{x}\,$ for some 1<a∈K{\textstyle 1<a\in K} in every formally exponential subfield K{\textstyle K} of the real numbers; however, $E({\sqrt {2}})=a^{\sqrt {2}}$ is not algebraic if 1<a{\textstyle 1<a} is algebraic by the Gelfond–Schneider theorem.
* Consequently, the class of formally exponential fields is not an elementary class since the field of real numbers and the field of real algebraic numbers are elementarily equivalent structures.
* The class of formally exponential fields is a pseudoelementary class. This is so since a field $K\,$ is exponentially closed if and only if there is a surjective function $E_2:K→K^+{\textstyle$ E_{2}\colon K\rightarrow K^{+}} such that $E_2(x+y)=E_2(x)E_2(y){\textstyle$ E_{2}(x+y)=E_{2}(x)E_{2}(y)} and $E_2(1)=2{\textstyle$ E_{2}(1)=2}; and these properties of $E_2{\textstyle$ E_{2}} are axiomatizable.
