In mathematics, an exponential field is a field with a further unary operation that is a homomorphism from the field's additive group to its multiplicative group. This generalizes the usual idea of exponentiation on the real numbers, where the base is a chosen positive real number.

Definition

A field is an algebraic structure and two binary operations forms an abelian group, one has

1. .

If there is also a function one has

. Thus an exponential function on a field is a homomorphism 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 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.,

Hence

.

Examples

• , has infinitely many exponential functions. One such function is the usual exponential function as required. Considering the ordered field.

• satisfies the required properties.

• Analogously to the real exponential field, there is the complex. However, this field is not ordered.

• Boris Zilber constructed an exponential field that, crucially, satisfies the equivalent formulation of Schanuel's conjecture with the field's exponential function., and a proof of this fact would thus prove Schanuel's conjecture.

Exponential rings

may not be required to be a field but instead allowed to simply be a ring to the multiplicative group of units. The resulting object is called an exponential ring.

that satisfy the conditions.

Open problems

Exponential fields are much-studied objects in model theory, occasionally providing a link between it and number theory as in the case of Zilber's work on Schanuel's conjecture is model complete, a result known as Wilkie's theorem. This result, when combined with Khovanskii's theorem on Pfaffian functions is also o-minimal. is not model complete. The question of decidability is still unresolved. Alfred Tarski and hence it is now known as Tarski's exponential function problem is decidable.

See also

• Ordered exponential field

Notes

References