Exponential field

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 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. Thus an exponential function on a field is a homomorphism between the additive group of F and 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. 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. The field of real numbers R, or (R, +, ·, 0, 1) as it may be written to highlight that we are considering it purely as a field with addition, multiplication, and special constants zero and one, has infinitely many exponential functions. One such function is the usual exponential function, that is E(x) = ex, since we have ex+y = exey and e0 = 1, as required. Considering the ordered field R equipped with this function gives the ordered real exponential field, denoted Rexp = (R, +, ·,