This elementary introduction offers a broad understanding of p adic numbers. More specifically, we define a new absolute value for any prime number p, by writing any rational number x in the form x p n ab where n is an integer and a and b are integers that are not divisible by p. Read online padic numbers, padic analysis, and zetafunctions pdf. The aim of the first chapter of this book is to introduce its main protagonist. Read download p adic numbers an introduction universitext. Slide 11 the p adics as a foundational tool hensel showed, in several papers and in his books, that one could. The solution of p adic equationsfrom 1943 published in the american mathe.
Padic numbers, padic analysis, and zetafunctions cep. 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. Kurt hensel1 around the year 1900 motivated by the idea of bringing the powerful tool of. Further, they offer a realm where one can dothings that are very similar to. Emory mathematics directed reading program program. This is the first book to deal with applications of p adic numbers in theoretical and. Those p adic numbers for which a i 0 for all i p adic integers, and form a subset of the p adic numbers commonly denoted. The isbns are 9783030472955 ebook and 9783030472948 softcover.
Sep 17, 2019 gouvea padic numbers pdf september 17, 2019 admin history leave a comment on gouvea padic numbers pdf to my eyes, the most glaring omission was a book by fernando gouvea, originally published in, entitled p adic numbers. 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. In this course, we will study p adic numbers, which play a central role. The p adic numbers are most simply a field extension of q, the rational. They also allow us to use methods from calculus and analysis to obtain results in algebra and number theory. Further, they offer a realm where one can dothings that are very similar to classical analysis, butwith results that are quite unusual. 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. Abstract algebra iii math 421422, real analysis i math 411. The p adic numbers are not as well known as the others, but they play a fundamental role in number theory and in other parts of mathematics, capturing information related to a chosen prime number p.
May 22, 2003 p adic numbers by fernando quadros gouvea, 9783540629115, available at book depository with free delivery worldwide. After doing this, one defines the p adic absolute value of x as x p p n. We want to construct an extension of q, q p, such that q. To my eyes, the most glaring omission was a book by fernando gouvea, originally published in, entitled p adic numbers. Dec 06, 2012 over the last century, padic numbers and p adic analysis have come to playa central role in modern number theory. Further, they offer a realm where one can do things that are very similar to classical analysis, but with results that are quite unusual. Pdf p adic numbers an introduction universitext download. The padic numbers are an extension of the field of rational. As opposed to real number expansions which extend to the right as sums of ever smaller, increasingly negative powers of the base p, p adic numbers may expand to the left forever, a property that. Then jaj 2 2, jaj 3 3 8, jaj 5 53, jaj p 1 for p 7. Aug 25, 2020 the p adic norm and the p adic numbers. 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. Pictures of ultrametric spaces, the padic numbers, and. Im especially interested in finding references to p adic numbers in papers published before ca.
The key prerequisite topics are groups, rings, elds, and. 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. The padic numbers are not as well known as the others, but they play a fundamental role in number theory and in other parts of mathematics, capturing information related to a chosen prime number p. This elementary introduction offers a broad understanding of padic numbers. Hensels analogy, how to compute, solving congruences modulo p n, other examples. An introduction universitext 2e from springer when. The p adic numbers were invented and introduced to number theory by. A joint publication of oxton house publishers, farmington, me. Read download p adic numbers an introduction universitext pdf. Further, they offer a realm where one can do things that are very similar to classical analysis, but with results that.
This paper introduces the padic numbers with an emphasis on comparison. An introduction, is a fun and conversational text with exercises sprinkled throughout. In other words, numbers that have many factors of p in the numerator are small and numbers with many factors of p in the denominator. In particular, two p adic numbers are considered to be close. May 22, 2003 over the last century, padic numbers and p adic analysis have come to playa central role in modern number theory.
Gouvea padic pdf posted on june 25, 2020 by admin to the padic number system as far as its metric and properties are concerned cation padic numbers. October 2001 ultrametric spaces, p adic numbers, valued fields 721. Lee corbin added it mar 10, one of the plans for the third edition is to include a little more on the history of the p adic numbers, and maybe something on witt vectors. The padic numbers are most simply a field extension of q, the rational. Intuitively, we might think of the distance between two species b and c as being a measure of the differences between their traits, the distance between b and c being small if they belong to the same genus, but large if they share only the same phylum. In particular, if and turn out to be natural numbers, then their sum as a p adic integer is no different from their sum as a natural number. The p adic integers will form a ring z p, which contains z. Pdf p adic numbers an introduction universitext download full.
It seems that hensels main motivation was the analogy between the ring of integers z, together with its. This series is called the p adic expansion of the number. Over the last century, padic numbers and p adic analysis have come to playa central role in modern number theory. Lately p adic numbers have attracted a great deal of attention in modern theoretical physics as a promising new approach for describing the nonarchimedean geometry of spacetime at small distances. The padic numbers were invented and introduced to number theory by. Jun 25, 2020 gouvea padic pdf posted on june 25, 2020 by admin to the padic number system as far as its metric and properties are concerned cation padic numbers. Mar 15, 2020 the p adic numbers were first p n for a, b, n. May 12, 2020 gouvea padic numbers pdf by admin posted on may 12, 2020 to my eyes, the most glaring omission was a book by fernando gouvea, originally published in, entitled p adic numbers. The p adic numbers are not as well known as the others, but they play a fundamental role in number theory and in other parts of mathematics. The p adic numbers are an extension of the field of rational. This is the first book to deal with applications of p adic numbers in theoretical. May 24, 2020 the p adic norm and the p adic numbers.
The extension is achieved by an alternative interpretation of the concept of closeness or absolute value. Nov 05, 2019 the p adic norm and the p adic numbers. Im especially interested in finding references to padic numbers in papers published before ca. It tries to be open up the theory to the reader in a friendly and accessible way while giving lots of hints about more advanced topics and applications. The p adic numbers are the set of all laurent expansions in p, i.
Proceedings of the third conference of the canadian number theory association held at queens university, kingston, ontario, august 1824, 1991. Check out the top books of the year on our page best books of thomas added it sep 27, books by fernando q. More generally, one can consider similar expansions in. Gouvea p adic numbers an introduction with 15 figures springerverlag berlin heidelberg gmbh fernando q. 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. It should be an enjoyable read for students with a wide range of experience. In addition to correcting all known typos, i added a lot of new material. The p adic rational numbers are then the completion of the rational numbers with respect to this absolute value. Number theory math 328 would help, but isnt required. The p adic numbers, where p is any prime number, come from an alternate way of defining. This is a common argument that is used in solving these types of equations.
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