Global field isomorphisms: a class field theoretical approach

Abstract

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field theory, we investigate two objects that uniquely determine the isomorphism type of a global field K, following an unpublished article of Cornelissen, Li, and Marcolli.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.