Inmathematics, anexponential field is afield with a further unary operation that is a homomorphism from the field's additive group to its multiplicative group. This generalizes the usual idea ofexponentiation on thereal numbers, where the base is a chosen positive real number.

A field is an algebraic structure composed of a set of elements,F, twobinary operations, addition (+) such thatF forms anabelian group with identity 0_F and multiplication (·), such thatF 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 elementsa,b,c inF, one hasa · (b +c) = (a ·b) + (a ·c). If there is also afunctionE that mapsF intoF, and such that for everya andb inF one has

thenF is called an exponential field, and the functionE is called an exponential function onF.^[1] Thus an exponential function on a field is ahomomorphism between the additive group ofF 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 havecharacteristic 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 elementx in a field with characteristicp > 0,

${\displaystyle 1=E(0)=E(\underset{p\text{ of these}}{\underbrace{x+x+\cdots +x}})=E(x)E(x)\cdots E(x)=E(x{)}^p.}$

Hence, taking into account theFrobenius endomorphism,

${\displaystyle (E(x)-1{)}^p=E(x{)}^p-1^p=E(x{)}^p-1=0.}$

And soE(x) = 1 for everyx.^[3]

• The field of real numbersR, 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 usualexponential function, that isE(x) =e^x, since we havee^x+y =e^xe^y ande⁰ = 1, as required. Considering theordered fieldR equipped with this function gives the ordered real exponential field, denotedR_exp = (R, +, ·, <, 0, 1, exp).
• Any real numbera > 0 gives an exponential function onR, where the mapE(x) =a^x satisfies the required properties.
• Analogously to the real exponential field, there is thecomplex exponential field,C_exp = (C, +, ·, 0, 1, exp).
• Boris Zilber constructed an exponential fieldK_exp that, crucially, satisfies the equivalent formulation ofSchanuel's conjecture with the field's exponential function.^[4] It is conjectured that this exponential field is actuallyC_exp, and a proof of this fact would thus prove Schanuel's conjecture.

The underlying setF may not be required to be a field but instead allowed to simply be aring,R, and concurrently the exponential function is relaxed to be a homomorphism from the additive group inR to the multiplicative group ofunits inR. The resulting object is called anexponential ring.^[2]

An example of an exponential ring with a nontrivial exponential function is the ring of integersZ equipped with the functionE which takes the value +1 at even integers and −1 at odd integers, i.e., the function${\displaystyle n\mapsto (-1{)}^n.}$ This exponential function, and the trivial one, are the only two functions onZ that satisfy the conditions.^[5]

Exponential fields are much-studied objects inmodel theory, occasionally providing a link between it andnumber theory as in the case ofZilber's work onSchanuel's conjecture. It was proved in the 1990s thatR_exp ismodel complete, a result known asWilkie's theorem. This result, when combined with Khovanskiĭ's theorem onpfaffian functions, proves thatR_exp is alsoo-minimal.^[6] On the other hand, it is known thatC_exp is not model complete.^[7] The question ofdecidability is still unresolved.Alfred Tarski posed the question of the decidability ofR_exp and hence it is now known asTarski's exponential function problem. It is known that if the real version of Schanuel's conjecture is true thenR_exp is decidable.^[8]

• Ordered exponential field

• Model theory
• Field (mathematics)
• Algebraic structures

• Articles with short description
• Short description is different from Wikidata