p-adic number

Abstract: 1 Aperitif.- 1 Aperitif.- 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.

Abstract: 1. NUMBERS: RATIONAL, REAL, p-ADIC We present a brief review of some selected topics in p-adic mathematical physics. More details can be found in the references below and the other references are mainly contained therein. We hope that this brief introduction to some aspects of p-adic mathematical physics could be helpful for the readers of the first issue of the journal p-Adic Numbers, Ultrametric Analysis and Applications. The notion of numbers is basic not only in mathematics but also in physics and entire science. Most of modern science is based on mathematical analysis over real and complex numbers. However, it is turned out that for exploring complex hierarchical systems it is sometimes more fruitful to use analysis over p-adic numbers and ultrametric spaces. p-Adic numbers (see, e.g. [1]), introduced by Hensel, are widely used in mathematics: in number theory, algebraic geometry, representation theory, algebraic and arithmetical dynamics, and cryptography.

Abstract: If X is a nonempty set, a distance, or metric, on X is a function d from pairs of elements (x, y) of X to the nonnegative real numbers such that $$ \begin{gathered} d(x,\,y) = 0\,if\,and\,only\,if\,x = y. \hfill \\ \hfill \\ \end{gathered} $$ (1) $$ d(x,\,y) = d(y,\,x). $$ (2) $$ d(x,\,y)\, \leqslant \,d(x,\,z) + d(z,\,y)\,for\,all\,z\, \in \,X. $$ (3) A set X together with a metrid d is called a metric space. The same set X can give rise to many different metric spaces (X, d), as we’ll soon see.

Abstract: Having built our foundation, we can now apply the general theory to the specific case of the field ℚ of rational numbers Extending our scope to include all fields of algebraic numbers (ie, finite extensions of ℚ), or even to include what the experts call “global fields” in general, would not be very hard Nevertheless, we have preferred to stick, at first, to the most concrete example available In a later chapter, we will consider some aspects of the problem of extending valuations from ℚ to larger fields More details about the theory of valuations on global fields can be found in several of the references