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

			Isomorphism <math>E\,</math> is called an exponential function in <math>F\,</math>.

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			* Every exponentially closed field <math>F\,</math> is ''root-closed'', i.e., every positive element of <math>F\,</math> has an <math>n\,</math>-th root for all positive integer <math>n\,</math> (or in other words the multiplicative group of positive elements of <math>F\,</math> is [[Divisible group\|divisible]]). This is so because <math>E\left(\frac{1}{n}E^{-1}(a)\right)^n=E(E^{-1}(a))=a</math> for all <math>a>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 number]]s is not exponentially closed. This is so because <math>E\,</math> has to be <math>E(x)=a^x\,</math> for some <math> 1<a\in F\,</math> in every exponentially closed subfield <math>F\,</math> of the real numbers; and <math>E(\sqrt{2})=a^\sqrt{2}</math> is not algebraic if <math> 1<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>F\,</math> is exponentially closed iff there is a surjective function <math>E_2:F\rightarrow F^+</math> such that <math>E_2(x+y)=E_2(x)E_2(y)\,</math> and <math>E_2(1)=2</math>; and these properties of <math>E_2\,</math> are axiomatizable.
			== See also ==
			* [[Ordered exponential field]]
			* [[Exponential field]]

			==References==
			{{cite journal \| first=Norman L. \| last=Alling \| title=On Exponentially Closed Fields \| journal=[[Proceedings of the American Mathematical Society]] \| volume=13 \| number=5 \| year=1962 \| pages=706–711 \| url=http://www.jstor.org/pss/2034159 \| zbl=0136.32201 }}

			[[Category:Model theory]]
			[[Category:Field theory]]
			[[Category:Algebraic structures]]

en>Yobot: WP:CHECKWIKI error fixes + general fixes, added orphan tag using AWB (7913)

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