Explicit class field theory: one dimensional and higher dimensional
This thesis investigates class field theory for one dimensional fields and higher dimensional fields. For one dimensional fields we cover the cases of local fields and global fields of positive characteristic. For higher dimensional fields we study the case of higher local fields of positive charact...
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| Format: | Thesis (University of Nottingham only) |
| Language: | English |
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2018
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| Online Access: | https://eprints.nottingham.ac.uk/50367/ |
| _version_ | 1848798233533349888 |
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| author | Yoon, Seok Ho |
| author_facet | Yoon, Seok Ho |
| author_sort | Yoon, Seok Ho |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | This thesis investigates class field theory for one dimensional fields and higher dimensional fields. For one dimensional fields we cover the cases of local fields and global fields of positive characteristic. For higher dimensional fields we study the case of higher local fields of positive characteristic. The main content of the thesis is divided into two parts.
The first part solves several problems directly related to Neukirch's axiomatic class field theory method. We first prove the famous Hilbert 90 Theorem in the case of tamely ramified extensions of local fields in an explicit way. This approach can be of use in understanding the role of the ring structure as opposed to the role of multiplication only in local class field theory. Next, we prove that for every local field, its `class field theory' is unique. Lastly, we establish the Neukirch axiom for global fields of positive characteristic, which leads to a new approach to class field theory for such fields, an approach that has not appeared in the previous literature.
There are two main successful directions in higher local class field theory, one by Kato and another by Fesenko. While Kato used a technical cohomological method, Fesenko generalised the Neukirch method and gave the first proof of the existence theorem. In the second part of the thesis we deal with the third method in class field theory that works in positive characteristic only, the Kawada-Satake method. We generalise the classical Kawada-Satake method to higher local fields of positive characteristic. We correct substantial mistakes in a paper of Parshin on such class field theory. We develop the first complete presentation of the theory based on the generalised Kawada-Satake method using advanced properties of topological Milnor K-groups. These advanced properties include Fesenko's theorem about relations of topological and algebraic properties of Milnor K-groups. |
| first_indexed | 2025-11-14T20:16:31Z |
| format | Thesis (University of Nottingham only) |
| id | nottingham-50367 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T20:16:31Z |
| publishDate | 2018 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-503672025-02-28T14:02:12Z https://eprints.nottingham.ac.uk/50367/ Explicit class field theory: one dimensional and higher dimensional Yoon, Seok Ho This thesis investigates class field theory for one dimensional fields and higher dimensional fields. For one dimensional fields we cover the cases of local fields and global fields of positive characteristic. For higher dimensional fields we study the case of higher local fields of positive characteristic. The main content of the thesis is divided into two parts. The first part solves several problems directly related to Neukirch's axiomatic class field theory method. We first prove the famous Hilbert 90 Theorem in the case of tamely ramified extensions of local fields in an explicit way. This approach can be of use in understanding the role of the ring structure as opposed to the role of multiplication only in local class field theory. Next, we prove that for every local field, its `class field theory' is unique. Lastly, we establish the Neukirch axiom for global fields of positive characteristic, which leads to a new approach to class field theory for such fields, an approach that has not appeared in the previous literature. There are two main successful directions in higher local class field theory, one by Kato and another by Fesenko. While Kato used a technical cohomological method, Fesenko generalised the Neukirch method and gave the first proof of the existence theorem. In the second part of the thesis we deal with the third method in class field theory that works in positive characteristic only, the Kawada-Satake method. We generalise the classical Kawada-Satake method to higher local fields of positive characteristic. We correct substantial mistakes in a paper of Parshin on such class field theory. We develop the first complete presentation of the theory based on the generalised Kawada-Satake method using advanced properties of topological Milnor K-groups. These advanced properties include Fesenko's theorem about relations of topological and algebraic properties of Milnor K-groups. 2018-07-19 Thesis (University of Nottingham only) NonPeerReviewed application/pdf en arr https://eprints.nottingham.ac.uk/50367/1/nottsthesis-test-full.pdf Yoon, Seok Ho (2018) Explicit class field theory: one dimensional and higher dimensional. PhD thesis, University of Nottingham. |
| spellingShingle | Yoon, Seok Ho Explicit class field theory: one dimensional and higher dimensional |
| title | Explicit class field theory: one dimensional and higher dimensional |
| title_full | Explicit class field theory: one dimensional and higher dimensional |
| title_fullStr | Explicit class field theory: one dimensional and higher dimensional |
| title_full_unstemmed | Explicit class field theory: one dimensional and higher dimensional |
| title_short | Explicit class field theory: one dimensional and higher dimensional |
| title_sort | explicit class field theory: one dimensional and higher dimensional |
| url | https://eprints.nottingham.ac.uk/50367/ |