Exponentially closed field

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

Isomorphism $E\,$ is called an exponential function in $F\,$.

Examples

• The canonical example for an exponentially closed field is the ordered field of real numbers; here $E\,$ can be any function $a^x\,$ where $1< a\in F$.

Properties

• Every exponentially closed field $F\,$ is root-closed, i.e., every positive element of $F\,$ has an $n\,$-th root for all positive integer $n\,$ (or in other words the multiplicative group of positive elements of $F\,$ is divisible). This is so because $E\left(\frac{1}{n}E^{-1}(a)\right)^n=E(E^{-1}(a))=a$ for all $a>0$.
• 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 $E\,$ has to be $E(x)=a^x\,$ for some $1<a\in F\,$ in every exponentially closed subfield $F\,$ of the real numbers; and $E(\sqrt{2})=a^\sqrt{2}$ is not algebraic if $1<a\,$ 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 $F\,$ is exponentially closed iff there is a surjective function $E_2:F\rightarrow F^+$ 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

• 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. Zbl 0136.32201.