#### Date of Award

2016

#### Document Type

Open Access Thesis

#### Department

Mathematics

#### Sub-Department

College of Arts and Sciences

#### First Advisor

Frank Thorne

#### Abstract

Let Vp denote the five dimensional vector space of binary quartic forms over the finite field Fp, with p a prime greater than 3. There is a natural action of the group GL1(Fp)×GL2(Fp) on Vp. This action partitions Vp into orbits, the number of which increases with p. In this thesis, we determine explicitly, for a given p, the number of orbits under the action of GL1(Fp) × GL2(Fp) on Vp. Moreover, we determine the size of each orbit and the general structure of the forms each orbit contains. We also introduce an application of understanding these orbits to the study of the Fourier transforms of certain functions over Vp that are of interest in algebraic number theory. We include two appendices with preliminary work towards extending key results from existing work on binary cubic forms to the case of binary quartic forms.

#### Recommended Citation

Kamenetsky, D. T.(2016). *Binary Quartic Forms over Fp.* (Master's thesis). Retrieved from https://scholarcommons.sc.edu/etd/3974