Girish Mahajan (Editor)

Exponentially closed field

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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 .

Contents

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 ∈ 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 ( 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 ∈ F in every exponentially closed subfield F of the real numbers; and E ( 2 ) = a 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 → 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

    Exponentially closed field Wikipedia
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