Definition
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 0F and multiplication (·), such that F excluding 0F forms an abelian group under multiplication with identity 1F, 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 between the additive group of F and its multiplicative group.
Trivial exponential function
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]
Exponential rings
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 
This exponential function, and the trivial one, are the only two functions on Z that satisfy the conditions. [5]
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