Dissertations, Theses, and Capstone Projects

Date of Degree

6-2024

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Gautam Chinta

Committee Members

Krzysztof Klosin

Andrew Obus

A. Raghuram

Subject Categories

Algebra | Number Theory

Keywords

higher composition laws, composition identities, Gauss composition, quadratic rings

Abstract

The theory of Gauss composition of integer binary quadratic forms provides a very useful way to compute the structure of ideal class groups in quadratic number fields. In addition to that, Gauss composition is also important in the problem of representations of integers by binary quadratic forms. In 2001, Bhargava discovered a new approach to Gauss composition which uses 2x2x2 integer cubes, and he proved a composition law for such cubes. Furthermore, from the higher composition law on cubes, he derived four new higher composition laws on the following spaces - 1) binary cubic forms, 2) pairs of binary quadratic forms, 3) pairs of quaternary alternating 2-forms, and 4) senary alternating 3-forms. All these spaces are naturally associated with the space of cubes. The class groups for these higher composition laws are all related to the narrow class group of quadratic rings. In fact, the class group structure, for each of the five spaces, is proved by establishing a bijection between orbits of the space under a natural group action and certain suitable ideal classes in quadratic rings. The aim of this thesis is to formulate these five higher composition laws in a manner similar to Gauss' formulation of composition of binary quadratic forms. We provide explicit composition identities for the higher composition laws and illustrate the identities via examples. These composition identities can be useful in understanding the properties of integers represented by these higher degree forms. We also highlight a few observations that serve to shed more light into the theory of higher composition laws.

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