Exponential Galois Theory (KIRBYJ_U26EMP)
Key Details
- Application deadline
- 31 March 2026. Project is open to Home applicants only.
- Location
- UEA
- Funding type
- Self-funded
- Start date
- 1 June 2026
- Mode of study
- Full-time
- Programme type
- Masters by Research
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Project description
Primary supervisor - Dr Jonathan Kirby
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 abelian. In the case of Zilber's field, this collection is exactly the real abelian numbers. The key ideas are that the real abelian numbers are related to values of the cosine function, and that the cosine function is defined from the complex exponential via a symmetric polynomial.
This project would build on that work, and upon more recent work done as an undergraduate summer project.
Entry requirements
The standard minimum entry requirement is 2:2 in Mathematics.
Funding
This project is offered on a self-funding basis. It is open to applicants with funding or those applying to funding sources. Details of tuition fees can be found here.
A bench fee is also payable in addition to the tuition fee to cover specialist equipment or laboratory costs required for the research. Applicants should contact the primary supervisor for further information about the fee associated with the project.
UEA Alumni 10% Scholarships - A scholarship of a 10% fee reduction is available to UEA Alumni looking to return for postgraduate study at UEA, Terms and conditions apply. For a postgraduate master’s loan, visit our Postgraduate Student Loans page for more information.
References
Kirby J, Macintyre A, Onshuus A. The algebraic numbers definable in various exponential fields. Journal of the Institute of Mathematics of Jussieu. 2012;11(4):825-834. doi:10.1017/S1474748012000047