Global field isomorphisms: a class field theoretical approach
Summary
This master’s thesis, Global field isomorphisms: a class field theoretical approach, was written
by Harry Smit from October 2015 until June 2016. It is submitted to the Department
of Mathematics at Utrecht University. The research was conducted under supervision
of professor Gunther Cornelissen, and the second reader is professor Frits Beukers.
After an introduction into both local and global class field theory, we investigate two
objects that uniquely determine the isomorphism type of a global field K, following
an unpublished article of Cornelissen, Li, and Marcolli. Firstly, we use the maximal
abelian Galois group to create a topological space X_K and subsequently a dynamical
system by defining an action of the integral ideals I_K on X_K. Secondly, we combine
the maximal abelian Galois group with the Dirichlet L-series. Both these objects
can be described using only objects from within K itself. The original contributions in
this thesis are the proof that X_K is a Hausdorff space and various improvements on
the proofs given by Cornelissen, Li, and Marcolli.