Ordered exponential field explained

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.^[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 $\backslash mathbf$ admits an exponential which extends the exponential function exp on R. Since $\backslash mathbf$ does not have the Archimedean property, this is an example of a non-Archimedean ordered exponential field.
• The ordered field of logarithmic-exponential transseries $\backslash mathbb^$ 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

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 $E\backslash left(\backslash fracE^(a)\backslash right)^n=E(E^(a))=a$ 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 E(\sqrt{ 2 }) = a ^{\sqrt{2}} is not algebraic if

• 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 $E\_2\backslash colon\; K\backslash rightarrow\; K^+$ 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

References

• Norman L. . Alling . On Exponentially Closed Fields . . 13 . 5 . 1962 . 706–711 . 0136.32201 . 10.2307/2034159. 2034159 . free .

Notes and References

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.