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...

وصف كامل

محفوظ في:
التفاصيل البيبلوغرافية
المؤلف الرئيسي: Song, Ruiqi
مؤلفون آخرون: Wu Guohua
التنسيق: Final Year Project
اللغة:English
منشور في: Nanyang Technological University 2023
الموضوعات:
الوصول للمادة أونلاين:https://hdl.handle.net/10356/166471
الوسوم: إضافة وسم
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المؤسسة: Nanyang Technological University
اللغة: English
الوصف
الملخص: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.