 Library Home /
 Search Collections /
 Open Collections /
 Browse Collections /
 UBC Theses and Dissertations /
 Taucongruent numbers and integers in number fields
Open Collections
UBC Theses and Dissertations
UBC Theses and Dissertations
Taucongruent numbers and integers in number fields Davis, Chad Tyler
Abstract
A positive integer n is the area of a Heron triangle if and only if there exists a nonzero rational number tau such that the elliptic curve E^n_tau: Y² = X(Xn tau)(X+n tau¹) has a rational point of order different than two. Such an integer n is called a taucongruent number. In the first part of this thesis, we give two distribution theorems on taucongruent numbers; in particular we show that given any fixed, nonzero rational tau, there exist infinitely many taucongruent numbers in every residue class modulo any positive integer m, and we also prove a similar result for taucongruent numbers whose corresponding elliptic curves have rank at least 2. In the second half of this thesis, we give some theorems on integers in certain algebraic number fields. First, we completely determine the index of a quartic field defined by a polynomial of the form X⁴ + aX + b. Second, we determine all possible normal integral bases in a certain cyclic quintic field. Finally, we give a family of cubic polynomials that define number fields that are both monogenic and have the same discriminant.
Item Metadata
Title 
Taucongruent numbers and integers in number fields

Creator  
Publisher 
University of British Columbia

Date Issued 
2019

Description 
A positive integer n is the area of a Heron triangle if and only if there exists a nonzero rational number tau such that the elliptic curve
E^n_tau: Y² = X(Xn tau)(X+n tau¹)
has a rational point of order different than two. Such an integer n is called a taucongruent number. In the first part of this thesis, we give two distribution theorems on taucongruent numbers; in particular we show that given any fixed, nonzero rational tau, there exist infinitely many taucongruent numbers in every residue class modulo any positive integer m, and we also prove a similar result for taucongruent numbers whose corresponding elliptic curves have rank at least 2.
In the second half of this thesis, we give some theorems on integers in certain algebraic number fields. First, we completely determine the index of a quartic field defined by a polynomial of the form X⁴ + aX + b. Second, we determine all possible normal integral bases in a certain cyclic quintic field. Finally, we give a family of cubic polynomials that define number fields that are both monogenic and have the same discriminant.

Genre  
Type  
Language 
eng

Date Available 
20190517

Provider 
Vancouver : University of British Columbia Library

Rights 
AttributionNonCommercialNoDerivatives 4.0 International

DOI 
10.14288/1.0378815

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Graduation Date 
201909

Campus  
Scholarly Level 
Graduate

Rights URI  
Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
Rights
AttributionNonCommercialNoDerivatives 4.0 International