On the finiteness of class groups

Abstract

In this thesis, We mainly discuss the finiteness of ideal class group for a number field and a function field of transcendence degree 1. The ideal class group is defined on Dedekind domains which is a measure of how close the Dedekind domain is to being a principal ideal domain. In a number field, the ideal class group is finite by using the embeddings which form specific absolute values on this number field. Our main goal in this thesis is to prove that the finiteness of ideal class group of a function field which is a finite extension of the one-variable rational function field with finite scalar field. To accomplish this goal, we review the proof of number fields case and construct the similar argument for function field case. The absolute values associated to the $\mathfrak{p}$-adic valuations are main tools. We give them the role which behaves like the absolute values associated to the embeddings as in the number field case. The proof of our main goal gives some clue for the cardinality of the ideal class group(the class number) for a function field. By using this, we compute the class number for some function field.