{"id":{"repo_id":"cape-town","oai_identifier":"oai:open.uct.ac.za:11427/32412"},"canonical_url":"https://search.dev.ndltd.org/etd/cape-town/oai:open.uct.ac.za:11427/32412","repository":{"repo_id":"cape-town","name":"University of Cape Town","base_url":"https://open.uct.ac.za/oai/request"},"display":{"title":"Extensive categories, commutative semirings and Galois theory","abstract":"We describe the Galois theory of commutative semirings as a Boolean Galois theory in the sense of Carboni and Janelidze. Such a Galois structure then naturally suggests an extension to commutative semirings of the classical theory of quadratic equations over commutative rings. We show, however, that our proposed generalization is impossible for connected commutative semirings which are not rings, leading to the conclusion that for the theory of quadratic equations, “minus is needed”. Finally, by considering semirings B which have no non-trivial additive inverses and no non-trivial zero divisors, we present an example of a normal extension of commutative semirings which has an underlying B-semimodule structure isomorphic to B×B.","abstract_html":"We describe the Galois theory of commutative semirings as a Boolean Galois theory in the sense of Carboni and Janelidze. Such a Galois structure then naturally suggests an extension to commutative semirings of the classical theory of quadratic equations over commutative rings. We show, however, that our proposed generalization is impossible for connected commutative semirings which are not rings, leading to the conclusion that for the theory of quadratic equations, “minus is needed”. Finally, by considering semirings B which have no non-trivial additive inverses and no non-trivial zero divisors, we present an example of a normal extension of commutative semirings which has an underlying B-semimodule structure isomorphic to B×B.","abstract_has_math":false,"creators":["Poklewski-Koziell, Rowan"],"institution":"Department of Mathematics and Applied Mathematics","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Janelidze, George"],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020","date_published":"2020","updated_at":"2026-07-22T22:23:06Z","subjects":["Applied Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/11427/32412","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Janelidze, George"]},{"key":"dc:creator","label":"Author","values":["Poklewski-Koziell, Rowan"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2020-11-19T12:07:05Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2020-11-19T12:07:05Z"]},{"key":"dc:date.issued","label":"Date","values":["2020"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["Department of Mathematics and Applied Mathematics"]},{"key":"dc:type","label":"Dc Type","values":["Master Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Masters","MSc"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Applied Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/11427/32412"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We describe the Galois theory of commutative semirings as a Boolean Galois theory in the sense of Carboni and Janelidze. Such a Galois structure then naturally suggests an extension to commutative semirings of the classical theory of quadratic equations over commutative rings. We show, however, that our proposed generalization is impossible for connected commutative semirings which are not rings, leading to the conclusion that for the theory of quadratic equations, “minus is needed”. Finally, by considering semirings B which have no non-trivial additive inverses and no non-trivial zero divisors, we present an example of a normal extension of commutative semirings which has an underlying B-semimodule structure isomorphic to B×B."]},{"key":"dc:title","label":"Title","values":["Extensive categories, commutative semirings and Galois theory"]}]}],"canonical_facts":{"dc:contributor.advisor":["Janelidze, George"],"dc:creator":["Poklewski-Koziell, Rowan"],"dc:date.accessioned":["2020-11-19T12:07:05Z"],"dc:date.available":["2020-11-19T12:07:05Z"],"dc:date.issued":["2020"],"dc:description.abstract":["We describe the Galois theory of commutative semirings as a Boolean Galois theory in the sense of Carboni and Janelidze. Such a Galois structure then naturally suggests an extension to commutative semirings of the classical theory of quadratic equations over commutative rings. We show, however, that our proposed generalization is impossible for connected commutative semirings which are not rings, leading to the conclusion that for the theory of quadratic equations, “minus is needed”. Finally, by considering semirings B which have no non-trivial additive inverses and no non-trivial zero divisors, we present an example of a normal extension of commutative semirings which has an underlying B-semimodule structure isomorphic to B×B."],"dc:identifier.uri":["http://hdl.handle.net/11427/32412"],"dc:publisher.department":["Department of Mathematics and Applied Mathematics"],"dc:subject":["Applied Mathematics"],"dc:title":["Extensive categories, commutative semirings and Galois theory"],"dc:type":["Master Thesis"],"dc:type.qualificationlevel":["Masters","MSc"]},"updated_at":"2026-07-22T22:23:06Z"}