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

Inhalt

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.