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 ${\textstyle E}$ on an ordered field ${\textstyle K}$ is a strictly increasing isomorphism of the additive group of ${\textstyle K}$ onto the multiplicative group of positive elements of ${\textstyle K}$. The ordered field ${\displaystyle K\,}$ together with the additional function ${\displaystyle E\,}$ 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 ${\textstyle a^{x}}$ where ${\textstyle a}$ 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.^[1] 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.^[2]
• The ordered field of surreal numbers ${\textstyle \mathbf {No} }$ admits an exponential which extends the exponential function exp on R. Since ${\textstyle \mathbf {No} }$ does not have the Archimedean property, this is an example of a non-Archimedean ordered exponential field.
• The ordered field of logarithmic-exponential transseries ${\textstyle \mathbb {T} ^{LE}}$ 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 ${\textstyle E}$. For any formally exponential field ${\textstyle K}$, one can choose an exponential ${\textstyle E}$ on ${\textstyle K}$ such that ${\textstyle 1+1/n<E(1)<n}$ for some natural number ${\textstyle n}$.^[3]

Properties

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

See also

• Exponential field

Notes

1. A.J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc., 9 (1996), pp. 1051–1094.
2. A.J. Macintyre, A.J. Wilkie, On the decidability of the real exponential field, Kreisel 70th Birthday Volume, (2005).
3. Salma Kuhlmann, Ordered Exponential Fields, Fields Institute Monographs, 12, (2000), p. 24.

References

• Alling, Norman L. (1962). "On Exponentially Closed Fields". Proceedings of the American Mathematical Society. 13 (5): 706–711. doi:10.2307/2034159. JSTOR 2034159. Zbl 0136.32201.
• Kuhlmann, Salma (2000), Ordered Exponential Fields, Fields Institute Monographs, vol. 12, American Mathematical Society, doi:10.1090/fim/012, ISBN 0-8218-0943-1, MR 1760173