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In mathematics, an exponential field is a field that has an extra operation on its elements which extends the usual idea of exponentiation.
A field is an algebraic structure composed of a set of elements, F, two binary operations, addition (+) such that F forms an abelian group with identity 0_{F} and multiplication (·), such that F excluding 0_{F} forms an abelian group under multiplication with identity 1_{F}, and such that multiplication is distributive over addition, that is for any elements a, b, c in F, one has a · (b + c) = (a · b) + (a · c). If there is also a function E that maps F into F, and such that for every a and b in F one has
then F is called an exponential field, and the function E is called an exponential function on F.^{[1]} Thus an exponential function on a field is a homomorphism from the additive group of F to its multiplicative group.
There is a trivial exponential function on any field, namely the map that sends every element to the identity element of the field under multiplication. Thus every field is trivially also an exponential field, so the cases of interest to mathematicians occur when the exponential function is non-trivial.
Exponential fields are sometimes required to have characteristic zero as the only exponential function on a field with nonzero characteristic is the trivial one.^{[2]} To see this first note that for any element x in a field with characteristic p > 0,
Hence, taking into account the Frobenius endomorphism,
And so E(x) = 1 for every x.^{[3]}
The underlying set F may not be required to be a field but instead allowed to simply be a ring, R, and concurrently the exponential function is relaxed to be a homomorphism from the additive group in R to the multiplicative group of units in R. The resulting object is called an exponential ring.^{[2]}
An example of an exponential ring with a nontrivial exponential function is the ring of integers Z equipped with the function E which takes the value +1 at even integers and −1 at odd integers, i.e., the function n \mapsto (-1)^n. This exponential function, and the trivial one, are the only two functions on Z that satisfy the conditions.^{[5]}
Exponential fields are a much-studied object in model theory, occasionally providing a link between it and number theory as in the case of Zilber's work on Schanuel's conjecture. It was proved in the 1990s that R_{exp} is model complete, a result known as Wilkie's theorem. This result, when combined with Khovanskiĭ's theorem on pfaffian functions, proves that R_{exp} is also o-minimal.^{[6]} On the other hand it is known that C_{exp} is not model complete.^{[7]} The question of decidability is still unresolved. Alfred Tarski posed the question of the decidability of R_{exp} 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 R_{exp} is decidable.^{[8]}
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