Exponentially closed field

In mathematics, an exponentially closed field is an ordered field of $F$ 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 $E{(\frac{1}{n}{E}^{-1}(a))}^n=E(E^{-1}(a))=a$ 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 $E_2:F\to F^{+}$ 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. https://www.jstor.org/pss/2034159.