Exponentially closed field

In mathematics, an exponentially closed field is an ordered field of [math]\displaystyle{ F\, }[/math] which has an order preserving isomorphism [math]\displaystyle{ E\, }[/math] of the additive group of [math]\displaystyle{ F\, }[/math] onto the multiplicative group of positive elements of [math]\displaystyle{ F\, }[/math] such that [math]\displaystyle{ 1+1/n\lt E(1)\lt n\, }[/math] for some natural number [math]\displaystyle{ n\, }[/math].

Isomorphism [math]\displaystyle{ E\, }[/math] is called an exponential function in [math]\displaystyle{ F\, }[/math].

Examples

• The canonical example for an exponentially closed field is the ordered field of real numbers; here [math]\displaystyle{ E\, }[/math] can be any function [math]\displaystyle{ a^x\, }[/math] where [math]\displaystyle{ 1\lt a\in F }[/math].

Properties

• Every exponentially closed field [math]\displaystyle{ F\, }[/math] is root-closed, i.e., every positive element of [math]\displaystyle{ F\, }[/math] has an [math]\displaystyle{ n\, }[/math]-th root for all positive integer [math]\displaystyle{ n\, }[/math] (or in other words the multiplicative group of positive elements of [math]\displaystyle{ F\, }[/math] is divisible). This is so because [math]\displaystyle{ E\left(\frac{1}{n}E^{-1}(a)\right)^n=E(E^{-1}(a))=a }[/math] for all [math]\displaystyle{ a\gt 0 }[/math].
• Consequently, every exponentially closed field is an Euclidean field.
• Consequently, every exponentially closed field is an ordered Pythagorean field.
• Not every real-closed field is an exponentially closed field, e.g., the field of real algebraic numbers is not exponentially closed. This is so because [math]\displaystyle{ E\, }[/math] has to be [math]\displaystyle{ E(x)=a^x\, }[/math] for some [math]\displaystyle{ 1\lt a\in F\, }[/math] in every exponentially closed subfield [math]\displaystyle{ F\, }[/math] of the real numbers; and [math]\displaystyle{ E(\sqrt{2})=a^\sqrt{2} }[/math] is not algebraic if [math]\displaystyle{ 1\lt a\, }[/math] is algebraic by Gelfond–Schneider theorem.
• Consequently, the class of exponentially closed 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 exponentially closed fields is a pseudoelementary class. This is so since a field [math]\displaystyle{ F\, }[/math] is exponentially closed iff there is a surjective function [math]\displaystyle{ E_2:F\rightarrow F^+ }[/math] such that [math]\displaystyle{ E_2(x+y)=E_2(x)E_2(y)\, }[/math] and [math]\displaystyle{ E_2(1)=2 }[/math]; and these properties of [math]\displaystyle{ E_2\, }[/math] are axiomatizable.

See also

• Ordered exponential field
• Exponential field

References

Alling, Norman L. (1962). "On Exponentially Closed Fields". Proceedings of the American Mathematical Society 13 (5): 706–711. doi:10.2307/2034159. https://www.jstor.org/pss/2034159.