On the Galois module structure of the units and ray classes of a real abelian number field

We study the Galois module structure of the ideal ray class group and the group of units of a real abelian number field. Specifically, we derive explicit annihilators of the ideal ray class groups in the vein of the classical Stickelberger theorems. This is made possible by generalizing a theorem of Rubin which in turn allows us to describe a relationship between the Galois module structure of certain explicit quotients of units and the Galois module structure of the ray class group. Along the way, we're compelled to study the Galois module structure of the p-adic completion of the units. We derive numerous conditions under which we may conclude that this module is cyclic some of which allow for p to divide the order of the Galois group. Under those conditions, we are able to relate the annihilators of the p-parts of various explicit quotients of units to annihilators of the p-parts of the ray class groups in many cases. This is a generalization of a theorem of Thaine.

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

doctoral

Discipline thesis:degree_discipline

Mathematics

Grantor dc:publisher

The Ohio State University

Year dc:date

2013

Author dc:creator

• All, Timothy James

Contributors dc:contributor

• Sinnott, Warren

• Mathematics
• real abelian number field
• class group
• Stickelberger
• units

Statement dc:rights

• unrestricted
• This thesis or dissertation is protected by copyright: all rights reserved. It may not be copied or redistributed beyond the terms of applicable copyright laws.

Language dc:language

English

Repository record dc:identifier

http://rave.ohiolink.edu/etdc/view?acc_num=osu1365594642

OAI identifier oai:identifier

oai:etd.ohiolink.edu:osu1365594642