Date of Award

1-1-2016

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

College/School/Department

Department of Mathematics and Statistics

Content Description

1 online resource (iii, 48 pages) : illustrations.

Dissertation/Thesis Chair

Anupam Srivastav

Committee Members

Marco Varisco, Boris Goldfarb, Alexandre Tchernev

Keywords

Kummer Theory, Locally Free Class Group, Normal Integral Basis, Number Thoery, Realisable Class, Swan Modules, Rings of integers, Finite groups, Group rings, Galois theory

Subject Categories

Physical Sciences and Mathematics

Abstract

Let $K$ be a number field and let $\mathcal{O}_{K}$ be its ring of integers. Let $L/K$ be a tame, cyclic, Kummer extension of number fields, whose Galois group is denoted by $G$. In this work we study the intersection of the Swan subgroup, $T(\mathcal{O}_{K}G)$, and the realisable subgroup, $R(\mathcal{O}_{K}G)$, of the class group, $Cl(\mathcal{O}_{K}G)$, of the group ring $\mathcal{O}_{K}G$. By modifying techniques of Del Corso and Rossi and extending previous results obtained by Replogle, we establish when a realisable class is in fact a Swan class by finding a normal integral basis of a \textit{twisted} ring $t\mathcal{O}_{L}+\mathcal{O}_{K}$, where $\mathcal{O}_{L}$ is the ring of integers of $L$ and $t$ is an algebraic integer relatively prime to deg$(L/K)$. We then present examples of degree 6 Galois extensions of $\Q (\zeta_3)$ and cyclic quartic extensions of $\Q(i)$ and moreover, show that $T(\mathcal{O}_KG)$ is nontrivial and contained in $R(\mathcal{O}_KG)$.

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