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p-adic Numbers

An Introduction. 'Universitext'. 2nd ed. 1997. Corr. 3r…
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Titel: p-adic Numbers
Autor/en: Fernando Quadros Gouvea

ISBN: 3540629114
EAN: 9783540629115
An Introduction.
2nd ed. 1997. Corr. 3rd printing 2003.
Sprache: Englisch.
Springer Berlin Heidelberg

22. Mai 2003 - kartoniert - 312 Seiten

There are numbers of all kinds: rational, real, complex, p-adic. The p-adic numbers are less well known than the others, but they play a fundamental role in number theory and in other parts of mathematics. This elementary introduction offers a broad understanding of p-adic numbers. From the reviews: "It is perhaps the most suitable text for beginners, and I shall definitely recommend it to anyone who asks me what a p-adic number is." --THE MATHEMATICAL GAZETTE
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.
From the reviews:

"This is a well-written introduction to the world of p-adic numbers. The reader is led into the rich structure of the fields Qp and Cp in a beautiful balance between analytic and algebraic aspects. The overall conclusion is simple: an extraordinarily nice manner to introduce the uninitiated to the subject. Not only giving the background necessary to pursue the matter, but doing it in such a way that a healthy 'hands-on experience'is generated in the process." Mededelingen van het wiskundig genootschap

"It is perhaps the most suitable text for beginners, and I shall definitely recommend it to anyone who asks me what a p-adic number is." The Mathematical Gazette

From the reviews of the second edition:

"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. ... Gouv ê a has succeeded admirably in taking a topic that is not standard in the undergraduate mathematics curriculum and writing a book accessible to undergraduates that allows its reader to play with some intriguing mathematics and explore a topic which is both fun and important." (Darren Glass, The Mathematical Association of America, January, 2011)

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