Exponentially Closed 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 $E$ on an ordered field $K$ is a strictly increasing isomorphism of the additive group of $K$ onto the multiplicative group of positive elements of $K$. 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 $a^x$ where $a$ is a real number greater than 1. One such function is the usual exponential function, that is . The ordered field R equipped with this function gives the ordered real exponential field, denoted by . 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 $\mathbf$ admits an exponential which extends the exponential function exp on R. Since $\mathbf$ does not have the Archimedean property, this is an example of a non-Archimedean ordered exponential field.
* The ordered field of logarithmic-exponential transseries $\mathbb^$ 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 $E$. For any formally exponential field $K$, one can choose an exponential $E$ on $K$ such that
1+1/n for some natural number $n$.Salma Kuhlmann, ''Ordered Exponential Fields'', Fields Institute Monographs, 12, (2000), p. 24.

Properties

* Every ordered exponential field $K$ is ''root-closed'', i.e., every positive element of $K\,$ has an $n$-th root for all positive integer $n$ (or in other words the multiplicative group of positive elements of $K\,$ is divisible). This is so because $E\left(\fracE^(a)\right)^n=E(E^(a))=a$ for all $a>0$.
* 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 $E$ has to be of the form $E(x)=a^x\,$ for some 1 in every formally exponential subfield $K$ of the real numbers; however, $E(\sqrt)=a^\sqrt$ is not algebraic if 1 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 $E_2\colon K\rightarrow K^+$ such that $E_2(x+y)=E_2(x)E_2(y)$ and $E_2(1)=2$; and these properties of $E_2$ are axiomatizable.

See also

* Exponential field

Notes

References

*
* {{Citation, last=Kuhlmann, first=Salma, title=Ordered Exponential Fields, series=Fields Institute Monographs, volume=12, publisher=American Mathematical Society, year=2000, isbn=0-8218-0943-1, mr=1760173, doi=10.1090/fim/012, doi-access=free

Model theory
Field (mathematics)
Algebraic structures
Exponentials