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.
1 Apéritif.- 2 Foundations.- 3 The p-adic Numbers.- 4 Exploring Qp.- 5 Elementary Analysis in Qp.- 6 Vector Spaces and Field Extensions.- 7 Analysis in Cp.- 8 Fun With Your New Head.- A Sage and GP: A (Very) Quick Introduction.- B Hints and Comments on the Problems.- C A Brief Glance at the Literature.- Bibliography.- Index.
Fernando Q. Gouvêa is a number theorist and historian of mathematics. In number theory he has been interested in the connection between p-adic modular forms and deformations of Galois representations. As a historian his main focus is on the early history of algebraic number theory, but he has also written on other historical topics. He enjoys books and has written many book reviews for a wide range of publications. For many years he served as editor of MAA Focus and of MAA Reviews. His current project is a forthcoming book on the history of p-adic numbers and p-adic analysis in the first decades of the twentieth century. Gouvêa’s other books include Arithmetic of p-adic Modular Forms, A Guide to Groups, Rings, and Fields, and (with William P. Berlinghoff) Math through the Ages: A Gentle History for Teachers and Others.
Dimensioni: 235 x 155 mm Ø 581 gr
Illustration Notes:19 Illustrations, black and white
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