Class groups of global fields

Abstract

In this thesis we study the class group of quadratic number fields and function fields over finite fields. Mostly we focus on the 2-primary part of the class group. Methods to calculate the 2-, 4-, and 8-rank of these class groups are explained for both number fields and function fields. The study of the 8-rank leads to a definition of Rédei symbols for function fields. Using a conditional theorem of Rédei reciprocity we prove the existence of governing fields for the 8-rank of the class group of a function field, analogous to a known version for number fields. Lastly we utilise the algebraic-geometric flavour of function fields over finite fields. By relating class groups to Jacobians of (hyper)elliptic curves over finite fields we find abelian groups not occurring as class groups, so-called missing class groups.