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.

References

Alling, Norman L. (1962). "On Exponentially Closed Fields". Proceedings of the American Mathematical Society. 13 (5): 706–711. Zbl 0136.32201. doi:10.2307/2034159.

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Exponentially closed field

Examples

Properties

See also

References