A Study On p-adic Analysis
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Date
2025
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Department of Mathematics, NITS
Abstract
This dissertation explores foundational and advanced topics in p-adic analysis, a field that extends classical analysis into the realm of p-adic numbers, which are constructed through the p-adic valuation and associated metric. The non-Archimedean nature of p-adic fields leads to unique analytical and topological behaviours that differ funda mentally from real and complex analysis. In this report, we have studied the p-adic systems and their relevance to broader mathematical structures. This report focuses on the p-adic numbers and some of the properties of them. We start by defining a norm and the p-adic absolute value and studied that it is a norm on the rational numbers. We also studied that the p-adic norm is a non-Archimedean norm which has strange properties compared to the ordinary (Archimedean) norm. Then we define the p-adic numbers and discuss that it is a complete field with respect to the p-adic norm.