Extensive categories, commutative semirings and Galois theory
| dc.contributor.advisor | Janelidze, George | |
| dc.contributor.author | Poklewski-Koziell, Rowan | |
| dc.date.accessioned | 2020-11-19T12:07:05Z | |
| dc.date.available | 2020-11-19T12:07:05Z | |
| dc.date.issued | 2020 | |
| dc.date.updated | 2020-11-19T08:42:00Z | |
| 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.apacitation | Poklewski-Koziell, R. (2020). <i>Extensive categories, commutative semirings and Galois theory</i>. (). ,Faculty of Science ,Department of Mathematics and Applied Mathematics. Retrieved from http://hdl.handle.net/11427/32412 | en_ZA |
| dc.identifier.chicagocitation | Poklewski-Koziell, Rowan. <i>"Extensive categories, commutative semirings and Galois theory."</i> ., ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 2020. http://hdl.handle.net/11427/32412 | en_ZA |
| dc.identifier.citation | Poklewski-Koziell, R. 2020. Extensive categories, commutative semirings and Galois theory. . ,Faculty of Science ,Department of Mathematics and Applied Mathematics. http://hdl.handle.net/11427/32412 | en_ZA |
| dc.identifier.ris | TY - Master Thesis AU - Poklewski-Koziell, Rowan AB - 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. DA - 2020_ DB - OpenUCT DP - University of Cape Town KW - Applied Mathematics LK - https://open.uct.ac.za PY - 2020 T1 - Extensive categories, commutative semirings and Galois theory TI - Extensive categories, commutative semirings and Galois theory UR - http://hdl.handle.net/11427/32412 ER - | en_ZA |
| dc.identifier.uri | http://hdl.handle.net/11427/32412 | |
| dc.identifier.vancouvercitation | Poklewski-Koziell R. Extensive categories, commutative semirings and Galois theory. []. ,Faculty of Science ,Department of Mathematics and Applied Mathematics, 2020 [cited yyyy month dd]. Available from: http://hdl.handle.net/11427/32412 | en_ZA |
| dc.language.rfc3066 | eng | |
| dc.publisher.department | Department of Mathematics and Applied Mathematics | |
| dc.publisher.faculty | Faculty of Science | |
| dc.subject | Applied Mathematics | |
| dc.title | Extensive categories, commutative semirings and Galois theory | |
| dc.type | Master Thesis | |
| dc.type.qualificationlevel | Masters | |
| dc.type.qualificationlevel | MSc |