# Exponentially closed field

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In mathematics, an **exponentially closed field** is an ordered field which has an order preserving isomorphism of the additive group of onto the multiplicative group of positive elements of such that
for some natural number .

Isomorphism is called an exponential function in .

## Examples

- The canonical example for an exponentially closed field is the ordered field of real numbers; here can be any function where .

## Properties

- Every exponentially closed field is
*root-closed*, i.e., every positive element of has an -th root for all positive integer (or in other words the multiplicative group of positive elements of is divisible). This is so because for all . - 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 has to be for some in every exponentially closed subfield of the real numbers; and is not algebraic if 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 is exponentially closed iff there is a surjective function such that and ; and these properties of are axiomatizable.

## See also

## References

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