Cryptography is defined to be the practice and studying of hiding information
and is used in applications present today; examples include the security of ATM
cards and computer passwords ([34]). In order to transform information to make it
unreadable, one needs a series of algorithms. Many of these algorithms are based on
elliptic curves because they require fewer bits. To use such algorithms, one must find
the rational points on an elliptic curve. The study of Algebraic Number Theory, and
in particular, rare objects known as power bases, help determine what these rational
points are. With such broad applications, studying power bases is an interesting
topic with many research opportunities, one of which is given below.
There are many similarities between Cyclic and Dihedral fields of prime degree;
more specifically, the structure of their field discriminants is comparable. Since the
existence of power bases (i.e. monogenicity) is based upon finding solutions to the
index form equation - an equation dependant on field discriminants - does this imply
monogenic properties of such fields are also analogous?
For instance, in [14], Marie-Nicole Gras has shown there is only one monogenic
cyclic field of degree 5. Is there a similar result for dihedral fields of degree 5? The
purpose of this thesis is to show that there exist infinitely many monogenic dihedral
quintic fields and hence, not just one or finitely many. We do so by using a well-
known family of quintic polynomials with Galois group D₅. Thus, the main theorem
given in this thesis will confirm that monogenic properties between cyclic and dihedral
quintic fields are not always correlative.
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:BVAU.2429/2788 |
| Date | 11 1900 |
| Creators | Lavallee, Melisa Jean |
| Publisher | University of British Columbia |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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