Valued fields: p-adic numbers as a case study

In this thesis, we first build the idea of field valuation and look into some of its examples, followed by the idea of general absolute value, as they are closely related. Then, an absolute value induces a metric, and we can talk about completions of fields with respect to the induced metric. In...

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Main Author: Song, Ruiqi
Other Authors: Wu Guohua
Format: Final Year Project
Language:English
Published: Nanyang Technological University 2023
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Online Access:https://hdl.handle.net/10356/166471
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Institution: Nanyang Technological University
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spelling sg-ntu-dr.10356-1664712023-05-08T15:38:56Z Valued fields: p-adic numbers as a case study Song, Ruiqi Wu Guohua School of Physical and Mathematical Sciences guohua@ntu.edu.sg Science::Mathematics::Algebra In this thesis, we first build the idea of field valuation and look into some of its examples, followed by the idea of general absolute value, as they are closely related. Then, an absolute value induces a metric, and we can talk about completions of fields with respect to the induced metric. In particular, we examine in detail the field of p-adic numbers as a great example. Finally, we return to general valued fields, and extract this process entirely onto the valuation level, bypassing absolute value and induced metric. Most importantly, we want to apply Hensel’s lemma on general valued fields, and in the end we shall see when it is possible. Bachelor of Science in Mathematical Sciences 2023-05-02T05:53:09Z 2023-05-02T05:53:09Z 2023 Final Year Project (FYP) Song, R. (2023). Valued fields: p-adic numbers as a case study. Final Year Project (FYP), Nanyang Technological University, Singapore. https://hdl.handle.net/10356/166471 https://hdl.handle.net/10356/166471 en application/pdf Nanyang Technological University
institution Nanyang Technological University
building NTU Library
continent Asia
country Singapore
Singapore
content_provider NTU Library
collection DR-NTU
language English
topic Science::Mathematics::Algebra
spellingShingle Science::Mathematics::Algebra
Song, Ruiqi
Valued fields: p-adic numbers as a case study
description In this thesis, we first build the idea of field valuation and look into some of its examples, followed by the idea of general absolute value, as they are closely related. Then, an absolute value induces a metric, and we can talk about completions of fields with respect to the induced metric. In particular, we examine in detail the field of p-adic numbers as a great example. Finally, we return to general valued fields, and extract this process entirely onto the valuation level, bypassing absolute value and induced metric. Most importantly, we want to apply Hensel’s lemma on general valued fields, and in the end we shall see when it is possible.
author2 Wu Guohua
author_facet Wu Guohua
Song, Ruiqi
format Final Year Project
author Song, Ruiqi
author_sort Song, Ruiqi
title Valued fields: p-adic numbers as a case study
title_short Valued fields: p-adic numbers as a case study
title_full Valued fields: p-adic numbers as a case study
title_fullStr Valued fields: p-adic numbers as a case study
title_full_unstemmed Valued fields: p-adic numbers as a case study
title_sort valued fields: p-adic numbers as a case study
publisher Nanyang Technological University
publishDate 2023
url https://hdl.handle.net/10356/166471
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