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Author Gouvรชa, Fernando Q. author p-adic Numbers [electronic resource] : An Introduction / by Fernando Q. Gouvรชa Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1997 Second Edition http://dx.doi.org/10.1007/978-3-642-59058-0 VI, 306 p. 1 illus. in color. online resource

SUMMARY

In the course of their undergraduate careers, most mathematics majors see little beyond "standard mathematics:" basic real and complex analysis, abยญ stract algebra, some differential geometry, etc. There are few adventures in other territories, and few opportunities to visit some of the more exotic corยญ 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 numยญ bers have shown up in other areas of mathematics, and even in physics

CONTENT

1 Apรฉritif -- 1 Apรฉritif -- 1.1 Henselโ{128}{153}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 Henselโ{128}{153}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

Mathematics Number theory Mathematics Number Theory

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