An exponential field is a field \(F\) of characteristic zero equipped with a homomorphism \(E\) from its additive group to its multiplicative group. The classical exponential fields are the real and complex fields. Zilber considered certain algebraically closed fields with surjective exponential map whose kernel is cyclic, as in the case of the complex field; following the paper under review, we call these fields Zilber fields. On top of the mentioned properties, these fields also satisfy the conclusion of Schanuel’s conjecture and they are strongly exponentially closed and have the countable closure property.
In this paper, the authors investigate the set of elements in certain exponential fields that are definable over the empty set in the language \(\{0,1,+,-,\cdot,E\}\) of exponential rings. First they prove that every real abelian number is definable in an exponential field with cyclic kernel and then that in the case of Zilber fields the real abelian numbers are exactly the definable elements. The field of real abelian numbers is defined as follows: the restrictions of all the involutions of \(\mathrm{Aut}(\overline{\mathbb Q})\) act in the same way on the maximal abelian extension \(\mathbb Q^{\text{ab}}\) of \(\mathbb Q\); then the field \(\mathbb Q^{\text{rab}}\) of real abelian numbers is simply the fixed field of this automorphism.
The proof of the first result is very elementary and neat. The authors use the fact that elements of \(\mathbb Q^{\text{ab}}\) can be expressed as certain exponential sums and they isolate the elements of \(\mathbb Q^{\text{rab}}\) as the rational combinations of certain special values of the cosine function (the definition of cosine function in the complex exponential field makes sense in any exponential field with cyclic kernel.)
The proof of the second result is much more involved and uses many facts known about the Zilber fields.