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＜電子ブック＞
p-adic Numbers : An Introduction / by Fernando Quadros Gouvea
(Universitext. ISSN:21916675)

版	2nd ed. 1997.
出版者	(Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer)
出版年	1997
大きさ	VI, 306 p. 1 illus. in color : online resource
著者標目	*Gouvea, Fernando Quadros author SpringerLink (Online service)
件　名	LCSH:Number theory FREE:Number Theory
一般注記	1 Apéritif -- 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 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 HTTP:URL=https://doi.org/10.1007/978-3-642-59058-0

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