Details

Subject(s)

• Number theory [Browse]

Series

• Universitext, [More in this series]
• Universitext, 2191-6675 [More in this series]

Summary note

In the course of their undergraduate careers, most mathematics majors see little beyond "standard mathematics:" basic real and complex analysis, ab­ stract algebra, some differential geometry, etc. There are few adventures in other territories, and few opportunities to visit some of the more exotic cor­ ners of mathematics. The goal of this book is to offer such an opportunity, by way of a visit to the p-adic universe. Such a visit offers a glimpse of a part of mathematics which is both important and fun, and which also is something of a meeting point between algebra and analysis. Over the last century, p-adic numbers and p-adic analysis have come to playa central role in modern number theory. This importance comes from the fact that they afford a natural and powerful language for talking about congruences between integers, and allow the use of methods borrowed from calculus and analysis for studying such problems. More recently, p-adic num­ bers have shown up in other areas of mathematics, and even in physics.

Notes

Bibliographic Level Mode of Issuance: Monograph

Source of description

Description based on publisher supplied metadata and other sources.

Language note

English

Contents

• 1 Apéritif
• 1.1 Hensel’s Analogy
• 1.2 Solving Congruences Modulopn
• 1.3 Other Examples
• 2 Foundations
• 2.1 Absolute Values on a Field
• 2.2 Basic Properties
• 2.3 Topology
• 2.4 Algebra
• 3 p-adic Numbers
• 3.1 Absolute Values on ?
• 3.2 Completions
• 3.3 Exploring ?p
• 3.4 Hensel’s Lemma
• 3.5 Local and Global
• 4 Elementary Analysis in ?p
• 4.1 Sequences and Series
• 4.2 Functions, Continuity, Derivatives
• 4.3 Power Series
• 4.4 Functions Defined by Power Series
• 4.5 Some Elementary Functions
• 4.6 Interpolation
• 5 Vector Spaces and Field Extensions
• 5.1 Normed Vector Spaces over Complete Valued Fields
• 5.2 Finite-dimensional Normed Vector Spaces
• 5.3 Finite Field Extensions
• 5.4 Properties of Finite Extensions
• 5.5 Analysis
• 5.6 Example: Adjoining a p-th Root of Unity
• 5.7 On to ?
• 6 Analysis in ?p
• 6.1 Almost Everything Extends
• 6.2 Deeper Results on Polynomials and Power Series
• 6.3 Entire Functions
• 6.4 Newton Polygons
• 6.5 Problems
• A Hints and Comments on the Problems
• B A Brief Glance at the Literature
• B.1 Texts
• B.2 Software
• B.3 Other Books.

Show 38 more Contents items

ISBN

3-642-59058-6

OCLC

927513702

Doi

10.1007/978-3-642-59058-0

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