Eichlinghofen, the 28th august 2015 by gilles bellot tudortmunduniversity. When we speak about padic numbers, p is some specific number, usually a prime, and the digits of the number are 0,p1. It is an extension of q, so the temptation is to imagine q. Gouvea my introduction to the padic numbers was last reprinted in 2003 by springerverlag.
In mathematics, particularly padic analysis, the padic exponential function is a padic analogue of the usual exponential function on the complex numbers. Using padic numbers, our theorem on quadratic forms then receives the following formulation its proof will appear in section 7. We will then consider fractions of padic integers, that is padic numbers, which we will show form the field qp. It took more than 25 years before p adic numbers were taken seriously by number theorists. More precisely, the padic numbers form the quotient field of the ring of padic integers. An introduction to the algorithmic of padic numbers. The absolute value p extends to qp by continuity q is dense in qp qp is complete with respect to the extended p. In contrast, the padic extension arises from the use of the counterintuitive. From image processing to topological modelling with padic numbers. Furthermore, if j j is a nonarchimedean value, then minfn 2 n. They have some things in common with usual numbers and they include the rational numbers math\mathbf qmath. The p in p adic is a variable and may be replaced with a prime yielding, for instance, the 2adic numbers or another placeholder variable for expressions such as the. We choose things to be certain ways to provide ourselves with a convenient language to describe and explore interesting phenomena.
Going from the rational numbers to the real numbers is one way to do that, but there are other ways namely, the padics for prime numbers p. Then a topology on that finite set of padic numbers yields a hierarchy of topological models underlying the image. Introduction to padic numbers an overview of ultrametric spaces and padic numbers. Real numbers are easy, since we see it as a line and lines are familiar, but padics are totally disconnects complete topological groups. Construction of padic numbers i let p be a prime, let an zpnz.
Urs hartl, a dictionary between fontainetheory and its analogue in equal characteristic arxiv. As in the complex case, it has an inverse function, named the padic logarithm definition. Recently, p adic numbers are keeping its importance to attract the mathemacians, and many other scientists with its applications in various areas. Koblitz, neal, p adic numbers, p adic analysis, and zetafunctions amazon, springerlink robert, alain m, a course in p adic analysis amazon, springerlink serre, jeanpierre, a course in arithmetic amazon, springerlink for me personally, learning general valuation theory was very useful for understanding p adic numbers. The p adic rational numbers are then the completion of the rational numbers with respect to this absolute value.
The field of padic numbers, qp, was first introduced by kurt hensel at the. We say a sequence in rn is cauchy for the iadic topology if for all t2n, there exists d2n such that whenever n. It seems that hensels main motivation was the analogy between the ring of integers z, together with its. Do they have any important applications outside of algebra that i could easily ex. A crucial difference is that the reals form an archimedean field, while the p padic numbers form a nonarchimedean field.
Pdf from image processing to topological modelling with padic. One way of looking at p adic numbers is that its a different way to fill in the gaps between rational numbers. Computations with padic numbers xavier caruso january 23, 2017 abstract this document contains the notes of a lecture i gave at the journ. The field of padic numbers is to the ring of padic integers what the field of rationals is to the ring of ordinary integers. We play a similar game to construct the padic numbers except that we choose a new notion of closeness. The synthetic differential geometryaspect of the p padic numbers is highlighted for instance in. In mathematics, 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 the real and complex number systems. Algebraic p adic expansions david lampert department of mathematics, universiry of michigan, ann arbor. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. Notice that rational numbers with large padic valuations have small padic absolute values.
We took advantage of this third printing of the second edition to correct a few typos. One way of looking at padic numbers is that its a different way to fill in the gaps between rational numbers. Nov 24, 2008 the padic numbers from the rational numbers. More precisely, the p adic numbers form the quotient field of the ring of p adic integers.
The p adic integers will form a ring z p, which contains z. If nis a natural number, and n a k 1 a k 2 a 1 a 0 is its padic representation in other wordsn p k. Lets try and nd a 10 adic number which is equal to 23. The aim of the lecture was to discuss lowlevel algorithmics for padic numbers. R q 1 the absolute value jj p extends to q p by continuity q is dense in q p q. Geometry of padic numbers zair ibragimov csuf valuations on rational numbers completions of rational numbers the padic numbers geometry of padic integers the standard valuation on q the real absolute value function jjon q is a wellknown example of a valuation. We study the padic deformation properties of algebraic cycle classes modulo rational equivalence. We show that the crystalline chern character of a vector bundle lies in a certain part of the hodge filtration if and only if, rationally, the class of the vector bundle lifts to a formal proclass in ktheory on the padic scheme. In algebra and number theory, one quite often deals with valuations instead of absolute values. That being said, the way they are made is not intuitive, as they encode arithmetic information rather than geometric information. Take an odd prime p, and an integer a coprime to p. Over the last century, p adic numbers and p adic analysis have come to playa central role in modern number theory. The purpose of this paper is to describe the padic numbers, but so far we have only established what the padic metric is.
The padic rational numbers are then the completion of the rational numbers with respect to this absolute value. In section 2 we introduce the notion of padic ring by analogy of the real case. What if its about p adic numbers or p adic analysis. Encoding the hierarchical structure of images by padic numbers allows for. With view to 2 one can decide whether an absolute value is archimedean or not. The p adic numbers are most simply a eld extension of q, the rational numbers, which can be formulated in two ways, using either analytic or algebraic methods. The latter in turn constitute an extension of the eld of rational numbers, analogous to the completion of the rationals by the real numbers with respect to the standard ordinary metric.
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. Defining a p adic number system defining a system of p adic numbers starts off being pretty similar to how we compute the representation of numbers in a standard numeric base. In mathematics, particularly p adic analysis, the p adic exponential function is a p adic analogue of the usual exponential function on the complex numbers. We will call padic digita natural number between 0 and p 1 inclusive. Hensel though they are foreshadowed in the work of his predecessor e. Thus in 2 only string worldsheet boundary x is treated. Before we can ask why q p is important, we must ask why these new metrics one for each prime p are important. Apr 29, 2012 this video is an exploratory video in which we loosely introduce an interesting variant on repeating decimals. The material presented in this chapter will be largely based upon that of gouvea 1 and baker 2. We show firstly, that it suffices to answer this for all prime powers.
Introduction to padic numbers an overview of ultrametric spaces and p adic numbers. Bernard le stum, one century of p padic geometry from hensel to berkovich and beyond talk notes, june 2012. 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. This is a common argument that is used in solving these types of equations. The field of p adic numbers for any prime p p adic numbers were invented in 1897 by kurt hensel 18611941. This addition of padic integers is associative, commutative, and veri. Meanwhile the theory of padic numbers has a plenty of applications and impacts in various mathematical. Subtraction of padic integers is also performed in exactly the same way as that of natural numbers in padic form. In particular, two padic numbers are considered to be close.
This video is an exploratory video in which we loosely introduce an interesting variant on repeating decimals. The aim of this talk is to give a survey on padic numbers and padic analysis. I thought about results on valuations of roots of polynomials, but it seems to me already too complicated par ailleurs since im introducing valuations at the beginning of the talk, it wont turn out to be an application to something they already knew. Pragpub april 20 padic math the pragmatic bookshelf. Computations with padic numbers archive ouverte hal. Worksheet on iadic topology and completion let rbe a ring and i. The adic of p adic comes from the ending found in words such as dyadic or triadic. Also many books and phd dissertations were published on p adic numbers and p adic analysis e. In padic terms, p100 is a very small number, and p is. Jul 23, 2017 in this video, we consider the question of solving a diophantine equation modulo all integers simultaneously.
The field of padic numbers for any prime p padic numbers were invented in 1897 by kurt hensel 18611941. 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. You can tell whether you have the new printing because this one has my photo on the back cover. However, in section 9 we prove a new result, namely a padic analogue. In p adic terms, p100 is a very small number, and p is even smaller. This elementary introduction offers a broad understanding of p adic numbers. Addition, subtraction, multiplication, division extend to the completionq p is a. Defining a padic number system defining a system of padic numbers starts off being pretty similar to how we compute the representation of numbers in a standard numeric base. The usual exponential function on c is defined by the. Over the last century, padic numbers and padic analysis have come to playa central role in modern number theory. Algebraic padic expansions david lampert department of mathematics, universiry of michigan, ann arbor. The field of p adic numbers is to the ring of p adic integers what the field of rationals is to the ring of ordinary integers. The field of padic numbers, qp, was first introduced by kurt hensel at the end of the 19th century.
As in the complex case, it has an inverse function, named the p adic logarithm. Ok, so padics are magical and wonderful number systems and it is my firm belief that everyone should love them as much as i do. Pictures of ultrametric spaces, the padic numbers, and. Fractions and padic numbers real numbers and limits math. Notice that rational numbers with large p adic valuations have small p adic absolute values. For people who prefer an algebraic approach, one can also define the padic integers as an inverse limit of the rings zp n z and then define the padic rational numbers as the fraction field of this inverse limit. This latter approach will allow to show that p adic integers form a ring, denoted by zp. Ill answer a simpler question, since the logic here is easier to see and extends to your question as well. We write this conventionally as a i a 2 a 1 a 0 that is, the a iare written from left to right.
Then a topology on that finite set of padic numbers yields a. Abstractencoding the hierarchical structure of images by padic numbers allows for image processing and. The p adic integers form a subset of the set of all p adic numbers. The padic integers form a subset of the set of all padic numbers. The extension is achieved by an alternative interpretation of the concept of closeness or absolute value. Important applications of padic numbers outside of algebra. For convenience, ill always take p 3, and as in class this case should su. Eichlinghofen, the 28th august 2015 by gilles bellot tudortmunduniversity facultyofmathematics. Going from the rational numbers to the real numbers is one way to do that, but there are other ways namely, the p adics for prime numbers p. Seventh international conference on p adic in conclusion, i recommend this book to the set of people interested in p adic numbers, and its complement. Harmonic analysis and representation theory of padic.