Harmonic analysis and representation theory of padic. For example, the equation can easily be shown to have no solutions in the field of 2 adic numbers. This text is suitable for an undergrad who has taken some analysis and topology. P adic numbers 5 ostrowski proved that any eld complete with respect to an archimedean absolute value is isomorphic to r or c. Kurt hensel 18611941 discovered the p adic numbers around the turn of the century. An introduction universitext kindle edition by fernando quadros gouvea. They have some things in common with usual numbers and they include the rational numbers math\mathbf qmath. Hensels lemma 10 acknowledgments references introduction the padics present a particularly rich subject. This is a wellwritten introduction to the world of padic numbers. For a fixed prime number p, a fraction of the form a p k, where a is a p adic integer and k is a nonnegative integer. If n d is a rational number in lowest terms, so that n and d are coprime, then is equal to if p divides n, or. An introduction to the algorithmic of padic numbers. In r, we may represent irrational numbers by nonterminating, nonrecurring decimal.
The goal of this book is to offer such an opportunity, by way of a visit. P adic analysis compared with real, by svetlana katok is a very gentle introduction to p adic numbers. Nov 24, 2008 the padic numbers from the rational numbers. This is a wellwritten introduction to the world of p adic numbers. The ring of p padic integers has the following properties. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields.
Information and translations of padic number in the most comprehensive dictionary definitions resource on the web. Hensel though they are foreshadowed in the work of his predecessor e. The padic valuation of 0 is defined to be infinity. Although for padic numbers p should be a prime, base 10 was chosen to highlight the analogy with decimals. In this paper i will survey the literature of padic model theory since. Though introductory, gouvea manages to incorporate some relatively advanced topics, as far as undergraduate mathematics goes such as the weierstrass. This book offers a selfcontained presentation of basic padic analysis. In mathematics, a padic zeta function, or more generally a padic lfunction, is a function analogous to the riemann zeta function, or more general lfunctions, but whose domain and target are padic where p is a prime number.
This elementary introduction offers a broad understanding of padic numbers. In particular, if and turn out to be natural numbers, then their sum as a padic integer is no different from their sum as a natural number. May 22, 2003 over the last century, p adic numbers and p adic analysis have come to playa central role in modern number theory. As a consequence, any eld that can be endowed with an archimedean absolute value is isomorphic to a sub eld of c. R q 1 the absolute value jj p extends to q p by continuity q is dense in q p q p is complete with respect to the extended jj p. It took more than 25 years before padic numbers were taken seriously by number theorists. Serre 1973 introduced padic modular forms as limits of ordinary modular forms, and katz 1973 shortly afterwards gave a geometric and more general definition. A course in p adic analysis, by alain robert is a more terse and advanced book on the subject.
On tdspaces, their smooth functions and distributions typical example of tdspace is the. This section is an informal introduction to padic numbers, using examples from the ring of 10adic decadic numbers. Padic analysis compared with real, by svetlana katok is a very gentle introduction to padic numbers. Padic numbers and bruhattits tree mcgill university. The main conjecture of iwasawa theory now a theorem due to barry mazur and andrew wiles is the statement that the kubotaleopoldt p adic l function and an arithmetic analogue constructed. On the other hand, there is a much larger variety of elds with a nonarchimedean absolute value.
Download it once and read it on your kindle device, pc, phones. We want to construct an extension of q, q p, such that q. I highly recommend ne chapter ii for a detailed discussion of this topic. In the course of their undergraduate careers, most mathematics majors see little beyond standard mathematics. Number theorists tend to emphasize the algebraic side and to focus on results that show up in their work. Koblitz padic numbers, padic analysis, and zetafunctions. Introduction to p adic numbers an overview of ultrametric spaces and p adic numbers. This section is an informal introduction to p adic numbers, using examples from the ring of 10 adic decadic numbers. This book offers a selfcontained presentation of basic p adic analysis. Besides we prove the existence of universal padic power series in the. Addition, subtraction, multiplication, division extend to the completionqp is a. The eld of p adic numbers q pis the fraction eld of z p. Preface to the second edition the most important revisions in this edition are.
Nicolas bourbaki, commutative algebra, hermann, 1972. Introduction the padic numbers were invented and introduced to number theory by. The reader is led into the rich structure of the fields qp and cp in a beautiful balance between analytic and algebraic aspects. Besides we prove the existence of universal p adic power series in the. Padic numbers 5 ostrowski proved that any eld complete with respect to an archimedean absolute value is isomorphic to r or c.
The padic numbers akhil mathew department of mathematics drew university math 155, professor alan candiotti 4 may 2009. The decadic numbers are generally not used in mathematics. For a fixed prime number p, a fraction of the form a p k, where a is a padic integer and k is a nonnegative integer. Functions, such as sqrt, log, exp, sin, cos, sinh, and cosh, are defined using. In this video, we consider the question of solving a diophantine equation modulo all integers simultaneously. Introduction to padic numbers an overview of ultrametric spaces and padic numbers. Eichlinghofen, the 28th august 2015 by gilles bellot tudortmunduniversity. Over the last century, padic numbers and padic analysis have come to playa. George bachman, introduction to padic numbers and valuation theory, academic press, 1964.
A padic lfunction arising in this way is typically called an arithmetic padic lfunction as it encodes arithmetic data of the galois module involved. Addition, subtraction, multiplication, division extend to the completionq p is a. The p adic rational numbers are then the completion of the rational numbers with respect to this absolute value. 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. We play a similar game to construct the padic numbers except that we choose a new notion of closeness. These exotic numbers or so they appeared at first are now wellestablished in the mathematical world and used more and more by physicists as well. Further, they offer a realm where one can do things that are very similar to classical analysis, but with results that are quite unusual. When we speak about padic numbers, p is some specific number, usually a prime, and the digits of the number are 0,p1. Kurt hensel 18611941 discovered the padic numbers around the turn of the century. It seems that hensels main motivation was the analogy between the ring of integers z, together with its.
Fractions and padic numbers real numbers and limits math foundations 90 n j wildberger duration. We give a brief and elementary introduction to p adic numbers and p adic functions. In mathematics, a p adic modular form is a p adic analog of a modular form, with coefficients that are p adic numbers rather than complex numbers. The padic numbers are less well known than the others, but they play a fundamental role in number theory and in other parts of mathematics. In this article we present a package that does padic calculations using mathematica. Calendar introduction to arithmetic geometry mathematics mit. In number theory, for a given prime number p, the padic order or padic valuation of a nonzero integer n is the highest exponent such that divides n. The adic numbers are useful in solving diophantine equations. It allows addition, subtraction, multiplication, and division of padic numbers written in hensel expansion. The representation function for sums of three squares along arithmetic progressions. We all know that p 2 is irrational, so that 2 is not a square in the rational eld q, but that we can enlarge q to the real eld r where 2 is a square.
Qp comes with a norm called the padic norm given by x qp p. Important applications of padic numbers outside of algebra. The absolute value p extends to qp by continuity q is dense in qp qp is complete with respect to the extended p. Dec 11, 2011 fractions and p adic numbers real numbers and limits math foundations 90 n j wildberger duration. The valuation of zp extend immediately to qp by letting vxy vx. The padic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to. 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. There are few adventures in other territories, and few opportunities to visit some of the more exotic cor ners of mathematics. In this video, we reinterpret padic numbers as power series. Historically, this research was inspired by the discovery that the veneziano amplitude of the open bosonic string, which is calculated using an integral over the real numbers, can be generalized to the padic numbers. The extensions of the padic numbers including extensions containing transcendentals, the correspondences between padic and real numbers, padic di. Wikimedia commons has media related to padic numbers. Although for p adic numbers p should be a prime, base 10 was chosen to highlight the analogy with decimals.
This elementary introduction offers a broad understanding of p adic numbers. A course in padic analysis, by alain robert is a more terse and advanced book on the subject. Do they have any important applications outside of algebra that i could easily ex. Just as the real numbers are the completion of the rationals with respect to the usual absolute valuation, the adic numbers are the completion of with respect to the adic valuation.
One checks that this does not depend on the choice of sequences representing x and y. It took more than 25 years before p adic numbers were taken seriously by number theorists. Historically, this research was inspired by the discovery that the veneziano amplitude of the open bosonic string, which is calculated using an integral over the real numbers, can be generalized to the p adic numbers. The eld of padic numbers q pis the fraction eld of z p. Recently, i was asked by the maa basic library list committee to look through the books on the maas basic library list, and to suggest any additions of books in number theory that might not already be on the list. Moreover, every point is an accumulation point, and there is a countable basis of clopen sets a stone space with these properties must be homeomorphic to cantor space. In mathematics, a padic modular form is a padic analog of a modular form, with coefficients that are padic numbers rather than complex numbers. Construction of padic numbers i let p be a prime, let an zpnz. For people who prefer an algebraic approach, one can also define the p adic integers as an inverse limit of the rings z p n z and then define the p adic rational numbers as the fraction field of this inverse limit. Books on p adic numbers and p adic analysis usually reflect quite closely the interests and concerns of their authors.
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