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Further material: Per WP:ISBN formatted 1 ISBNs using AWB (7596)

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In [[mathematics]], an '''exponential field''' is a [[Field (mathematics)|field]] that has an extra operation on its elements which extends the usual idea of [[exponentiation]].

==Definition==

A field is an algebraic structure composed of a set of elements, ''F'', and two operations, addition ('+') and multiplication ('·'), such that the set of elements forms an [[abelian group]] under both operations with identities 0''F'' and 1''F'' respectively, 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 (mathematics)|function]] ''E'' that maps ''F'' into ''F'', and such that for every ''a'' and ''b'' in ''F'' one has
:<math>\begin{align}&E(a+b)=E(a)\cdot E(b),\\
&E(0_F)=1_F \end{align}</math>
then ''F'' is called an exponential field, and the function ''E'' is called an exponential function on ''F''.<ref>Helmut Wolter, ''Some results about exponential fields (survey)'', Mémoires de la S.M.F. 2e série, '''16''', (1984), pp.85–94.</ref> Thus an exponential function on a field is a [[homomorphism]] from the additive group of ''F'' to 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 (algebra)|characteristic]] zero as the only exponential function on a field with nonzero characteristic is the trivial one.<ref name="Dries">Lou van den Dries, ''Exponential rings, exponential polynomials and exponential functions'', Pacific Journal of Mathematics, '''113''', no.1 (1984), pp.51–66.</ref> To see this first note that for any element ''x'' in a field with characteristic ''p'' > 0,
:<math>1=E(0)=E(\underbrace{x+x+\ldots+x}_{p\text{ of these}})=E(x)E(x)\cdots E(x)=E(x)^p.</math>
Hence, taking into account the [[Frobenius endomorphism]],
:<math>(E(x)-1)^p=E(x)^p-1^p=E(x)^p-1=0.\,</math>
And so ''E''(''x'') = 1 for every ''x''.<ref>Martin Bays, Jonathan Kirby, A.J. Wilkie, ''A Schanuel property for exponentially transcendental powers'', (2008), {{arxiv|0810.4457}}</ref>

==Examples==

* 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'') = [[e (mathematical constant)|''e'']]''x'', since we have ''e''''x''+''y'' = ''e''''x''''e''''y'' and ''e''0 = 1, as required. Considering the [[ordered field]] '''R''' equipped with this function gives the ordered real exponential field, denoted '''R'''exp = ('''R''',+,·,<,0,1,exp).

* In fact any real number ''a'' > 0 gives an exponential function on '''R''', specifically the map ''E''(''x'') = ''a''''x'' satisfies the required properties.

* Analogously to the real exponential field, there is the [[Complex number|complex]] exponential field, '''C'''exp = ('''C''',+,·,0,1,exp).

* Boris Zilber constructed an exponential field ''K''exp that, crucially, satisfies the equivalent formulation of [[Schanuel's conjecture]] with the field's exponential function.<ref>Boris Zilber, ''Pseudo-exponentiation on algebraically closed fields of characteristic zero'', Ann. Pure Appl. Logic, '''132''', no.1 (2005), pp.67–95.</ref> It is conjectured that this exponential field is actually '''C'''exp, and a proof of this fact would thus prove Schanuel's conjecture.

==Exponential rings==
The underlying set ''F'' may not be required to be a field but instead allowed to simply be a [[Ring (mathematics)|ring]], ''R'', and concurrently the exponential function is relaxed to be a homomorphism from the additive group in ''R'' to the multiplicative group of [[Unit (ring theory)|units]] in ''R''. The resulting object is called an '''exponential ring'''.<ref name="Dries"/>

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 <math>n \mapsto (-1)^n.</math> This exponential function, and the trivial one, are the only two functions on '''Z''' that satisfy the conditions.<ref>Giuseppina Terzo, ''Some Consequences of Schanuel's Conjecture in Exponential Rings'', Communications in Algebra, Volume 36, Issue 3 (2008), pp.1171–1189.</ref>

==Open problems==

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 theory|model complete]], a result known as [[Wilkie's theorem]]. This result, when combined with Khovanskiĭ's theorem on [[pfaffian function]]s, proves that '''R'''exp is also [[o-minimal]].<ref>A.J. Wilkie, ''Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function'', J. Amer. Math. Soc., '''9''' (1996), pp.1051–1094.</ref> On the other hand it is known that '''C'''exp is not model complete.<ref>David Marker, ''A remark on Zilber's pseudoexponentiation'', The Journal of Symbolic Logic, '''71''', no.3 (2006), pp.791–798.</ref> The question of [[Decidability (logic)|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.<ref>A.J. Macintyre, A.J. Wilkie, ''On the decidability of the real exponential field'', Kreisel 70th Birthday Volume, (2005).</ref>

==See also ==
* [[Ordered exponential field]]
* [[Exponentially closed field]]

==Notes==
{{Reflist}}

[[Category:Model theory]]
[[Category:Field theory]]
[[Category:Algebraic structures]]

Elliptic divisibility sequence - Revision history

en>RjwilmsiBot: /* Further material */Per WP:ISBN formatted 1 ISBNs using AWB (7596)