Exponential Galois Theory (KIRBYJ_U26EMP)

The solutions of a polynomial equation have an algebraic structure, a Galois group, which gives the symmetry group of the solutions under field automorphisms. This project will look at the analogous situation but where the field is equipped with an exponential map. Automorphisms have to fix the exponential, so there are fewer automorphisms, and this is particularly clear for the roots of unity, since they are all of the form exp(πiq) for a rational q, so once we fix πi to be itself or its complex conjugate, there are no more automorphisms!

However as well as solutions to polynomial equations, we can also ask about the exponential Galois groups of solutions to exponential equations. For example, the Galois group of log(2), the solution to exp(x)=2, is the infinite group Z of integers under addition. Two settings are of interest: the complex exponential field and Zilber's exponential field.

The first is hard to work with because we do not know the answers to relevant questions, such as whether e and π are algebraically independent. Zilber's field has the answers to these questions built in, and is much more accessible for exact calculations. A 2012 paper by Kirby, Macintyre and Onshuus explains some of the background.

There, the authors showed that the collection of algebraic numbers which are fixed by all exponential automorphisms ("exponential rational numbers") contains the real abelian algebraic numbers, that is, all those numbers which are real and whose Galois group is a