Cyclotomic units and stickelberger ideals over global function fields

Abstract

In this thesis we study on the cyclotomic untis and the Stickelberger ideal of abelian extensions F/k, which are subfields of cyclotomic function fields over a global function field k. In chapter 1, we give a brief history about cyclotomic units and Stickelber ideals in number theory. In chapter 2, we give the theory of cyclotomic function fields briefly. We also give definitions and properties of the logarithm map, the restriction and corestriction map, the lattice index, which will be used throughout the paper. In chapter 3, we define the group of cyclotomic units $C_F$ and we calculate the index of $C_F$ in $O_F^*$, which involves the ideal class number of its maximal real subfield. In chapter 4, we define the Stickelberger ideals $I_{F}^{±}$ and $I_{F}$ and we calculate their indices $[R^{±} : I_{F}^{±}]$ and $[R : I_{F}]$, whose formulas involve the class numbers $h^{-}(O_{F})$, $h(F^+)$ and h(F), respectively. In chapter 5, we discuss on the indices $(R : U), (e^{+}R : e^{+}U)$ and $(e^{-}R : e^{-}U)$ which appear in our index-class number formulas and calculate them for some cases.