Relative oriented class groups of quadratic extensions

In 2018 Zemková defined relative oriented class groups associated to quadratic extensions of number fields L/K, extending work of Bhargava concerning composition laws for binary quadratic forms over number fields of higher degree. This work generalized the classical correspondence between ideal classes of quadratic orders and classes of integral binary quadratic forms to any base number field of narrow class number 1. Zemková explicitly computed these relative oriented class groups for quadratic extensions of the rationals. We consider extended versions of this work and develop general strategies to compute relative oriented class groups for quadratic extensions of higher degree number fields by way of the action of Gal(K/Q) on the set of real embeddings of K. We also investigate the binary quadratic forms side of Zemková's bijection and determine conditions for representability of elements of K. Another project comprising work done jointly with Lian Duan, Ning Ma, and Xiyuan Wang is included in this thesis. Our project investigates a principal version of the Chebotarev density theorem, a famous theorem in algebraic number theory which describes the splitting of primes in number field extensions. We provide an overview of the formulation of the principal density and describe its connection to the splitting behavior of the Hilbert exact sequence.

Name thesis:degree_name

Doctor of Philosophy (Ph.D.)

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Mathematics

Grantor dc:publisher

Colorado State University. Libraries

Year dc:date.issued

2024

Authors dc:creator

• O'Connor, Kelly A., author
• Pries, Rachel, advisor
• Achter, Jeffrey, committee member
• Shoemaker, Mark, committee member
• Rugenstein, Maria, committee member

Statement dc:rights

• Copyright and other restrictions may apply. User is responsible for compliance with all applicable laws. For information about copyright law, please see https://libguides.colostate.edu/copyright.

Language dc:language.iso

eng, English

Identifier URI

https://doi.org/10.25675/3.02359

OAI identifier oai:identifier

oai:mountainscholar.org:10217/238477