The canonical example for an ordered exponential field is the ordered field of real numbers R with any function of the form where is a real numbergreater than 1. One such function is the usual exponential function, that is. The ordered field R equipped with this function gives the ordered real exponential field, denoted by. It was proved in the 1990s that Rexp is model complete, a result known as Wilkie's theorem. This result, when combined with Khovanskiĭ's theorem on pfaffian functions, proves that Rexp is also o-minimal. Alfred Tarski posed the question of the decidability of Rexp and hence it is now known as Tarski's exponential function problem. It is known that if the real version of Schanuel's conjecture is true then Rexp is decidable.
The ordered field of surreal numbers admits an exponential which extends the exponential function exp on R. Since does not have the Archimedean property, this is an example of a non-Archimedean ordered exponential field.
A formally exponential field, also called an exponentially closed field, is an ordered field that can be equipped with an exponential. For any formally exponential field, one can choose an exponential on such that for some natural number.
Properties
Every ordered exponential field is root-closed, i.e., every positive element of has an -th root for all positive integer . This is so because for all.
Consequently, every ordered exponential field is a Euclidean field.
Consequently, every ordered exponential field is an ordered Pythagorean field.
Not every real-closed field is a formally exponential field, e.g., the field of real algebraic numbers does not admit an exponential. This is so because an exponential has to be of the form for some in every formally exponential subfield of the real numbers; however, is not algebraic if is algebraic by the Gelfond–Schneider theorem.
Consequently, the class of formally exponential 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 formally exponential fields is a pseudoelementary class. This is so since a field is exponentially closed if and only if there is a surjective function such that and ; and these properties of are axiomatizable.
