p-adic Analysis Compared with Real

The courses of p-adic (non-Archimedean) analysis seldom find a place in the undergraduate curriculum and often run separately from the classical (Archimedean) one with sporadic references to mutual differences. This book tries to introduce the reader to the first one making use of his/her familiarity with the classical counterpart and an intuitive experience from number theory and topology. The book starts with the construction of p-adic numbers based on the completion process of the field of rationals by replacing the Euclidean distance by a p-adic one. Then it continues with a study of their basic arithmetic and algebraic properties and a comparison of the topology of the field p-adic numbers, where p is a prime, with the field of real numbers (e.g. the strong triangle inequality, that balls are clopen, and the condition for the convergence of series).

The analytic highlights of the book are formed by elements of the analysis in the field of p-adic numbers and the basics of the calculus of p-adic functions. The reader will also find here some non-standard topics for undergraduate courses, including totally disconnected spaces, the Baire Category Theorem, and isometries of compact metric spaces. There are a large number of useful exercises and this makes the book more readable for advanced undergraduate or postgraduate students who want to learn something about p-adic numbers. In fact, many of them extend or complement the presented theory and some of them complete the proofs of results appearing in the text. Due to the role of p-adic numbers in many branches of mathematics and physics the book gives a good impetus to students to study the “p-adic worlds” more deeply. This role of the book is not only supported by carefully selected material but also by the fact that it is written in a very lively and lucid style.

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