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.
","abstract_html":"<p>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.</p>","abstract_has_math":false,"creators":["Nair, Ajith A"],"institution":"The Graduate School and University Center of The City University of New York","degree_name":"Doctor of Philosophy","degree_level":"Doctoral","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":["Gautam Chinta"],"committee_chairs":[],"committee_members":["Krzysztof Klosin","Andrew Obus","A. Raghuram"],"year":2024,"date_issued":"2024-06-01T07:00:00Z","date_published":"2024-06-01T07:00:00Z","updated_at":"2026-07-24T01:58:39Z","subjects":["Algebra","Number Theory","higher composition laws","composition identities","Gauss composition","quadratic rings"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://academicworks.cuny.edu/gc_etds/5838","outbound_label":"Repository record","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Gautam Chinta"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Krzysztof Klosin","Andrew Obus","A. 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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.
"]},{"key":"dc:title","label":"Title","values":["Explicit Composition Identities for Higher Composition Laws in the Quadratic Case"]}]}],"canonical_facts":{"dc:contributor.advisor":["Gautam Chinta"],"dc:contributor.committeemember":["Krzysztof Klosin","Andrew Obus","A. Raghuram"],"dc:creator":["Nair, Ajith A"],"dc:date.available":["2024-04-30T07:00:00Z"],"dc:description.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.
"],"dc:identifier":["https://academicworks.cuny.edu/gc_etds/5838"],"dc:subject":["Algebra","Number Theory","higher composition laws","composition identities","Gauss composition","quadratic rings"],"dc:title":["Explicit Composition Identities for Higher Composition Laws in the Quadratic Case"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Doctoral"],"thesis:degree_name":["Doctor of Philosophy"],"thesis:institution_name":["The Graduate School and University Center of The City University of New York"]},"updated_at":"2026-07-24T01:58:39Z"}