Fernando Q. Further, they offer a realm where one can do things that are very similar to classical analysis, but with results that are quite unusual. The book should be of use to students interested in number theory, but at the same time offers an interesting example of the many connections between different parts of mathematics. The book strives to be understandable to an undergraduate audience. Very little background has been assumed, and the presentation is leisurely.
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Fernando Q. Further, they offer a realm where one can do things that are very similar to classical analysis, but with results that are quite unusual. The book should be of use to students interested in number theory, but at the same time offers an interesting example of the many connections between different parts of mathematics. The book strives to be understandable to an undergraduate audience.
Very little background has been assumed, and the presentation is leisurely. There are many problems, which should help readers who are working on their own a large appendix with hints on the problem is included. Most of all, the book should offer undergraduates exposure to some interesting mathematics which is off the beaten track. Those who will later specialize in number theory, algebraic geometry, and related subjects will benefit more directly, but all mathematics students can enjoy the book.
Elementary Analysis in Qp Vector Spaces and Field Extensions. Analysis in Co. A Hints and Comments on the Problems B A Brief Glance at the Literature. P-Adic Numbers Fernando Q. Foundations
Fernando Gouvea. 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.
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries. My own first exposure to the book came when I was an undergraduate and one of my friends came back from an REU and gave a talk on p-adic numbers. I was intrigued by one of the counterintuitive results that one gets when working in the p-adic topology — all triangles are isosceles, but none are equilateral! I checked it out of the library and found a book that was mathematically rich while still being clear and lively to read. In the intervening decade and a half, as I have spent much time reading about and working with p-adic numbers in my own research, I have read any number of treatments of the subject, and certainly several other sources come to mind that have various advantages in their different goals and perspectives. But if I had to recommend one book on the subject to a student — or even to a fully grown mathematician who had never played with p-adic numbers before — it would still be this book. It turns out that the p-adic rational numbers are similar to the real numbers in some senses they are locally compact completions of the normal rational numbers, they are not algebraically closed while being very different in other senses the p-adic rationals are totally disconnected and are not ordered, for example.