PirateBay Download | hatcher topology numbers2022

hatcher topology numbers2022

Type:: E-books
Files:: 1
Size:: 19.0 MB
Uploaded On:: March 11, 2022, 7:03 a.m.
Added By:: andryold1
Seeders:: 4
Leechers:: 0
Info Hash:: 44135297B4848FAE4191BB331AB389FAA7833CF8

Textbook in PDF format This book provides an introduction to Number Theory from a point of view that is more geometric than is usual for the subject, inspired by the idea that pictures are often a great aid to understanding. The title of the book, Topology of Numbers, is intended to express this visual slant, where we are using the term “Topology" with its general meaning of “the spatial arrangement and interlinking of the components of a system". The other unusual aspect of the book is that, rather than giving a broad introduction to all the basic tools of Number Theory without going into much depth on any one, it concentrates largely on a single topic, quadratic forms Q (x, y) = ax2 + bxy + cy2 with integer coeﬃcients, where there is a very rich theory that one can really immerse oneself in to get a real feeling for the beauty and subtlety of Number Theory. Along the way we do in fact encounter many standard number-theoretic tools, but with a better idea of their usefulness. A central geometric theme of the book is a certain two-dimensional ﬁgure known as the Farey diagram, discovered by Adolf Hurwitz in 1894, which displays certain relationships between rational numbers beyond just their usual distribution along the one-dimensional real number line. Among the many things the diagram elucidates that will be explored in the book are Pythagorean triples, the Euclidean algorithm, Pell’s equation, continued fractions, Farey sequences, and two-by-two matrices with integer entries and determinant ±1. But most importantly for this book, the Farey diagram can be used to study quadratic forms Q(x, y) = ax2 + bxy + cy2 in two variables with integer coeﬃcients via John Conway’s marvelous idea of the topograph of such a form. The origins of the wonderfully subtle theory of quadratic forms can be traced back to ancient times. In the 1600s interest was reawakened by numerous discoveries of Fermat, but it was only in the period 1750-1800 that Euler, Lagrange, Legendre, and especially Gauss were able to uncover the main features of the theory. The principal goal of the book is to present an accessible introduction to this theory from a geometric viewpoint that complements the usual purely algebraic approach. Prerequisites for reading the book are fairly minimal, hardly going beyond high school mathematics for the most part. One topic that often forms a signiﬁcant part of elementary number theory courses is congruences modulo an integer n. It would be helpful if the reader has already seen and used these a little, but we will not develop congruence theory as a separate topic and will instead just use congruences as the need arises, proving whatever nontrivial facts are required including several of the basic ones that form part of a standard introductory number theory course. Among these is quadratic reciprocity, where we give Eisenstein’s classical proof since it involves some geometry. The high point of the basic theory of quadratic forms Q (x, y) is the class group ﬁrst constructed by Gauss. This can be deﬁned purely in terms of quadratic forms, which is how it was ﬁrst presented, or by means of Kronecker’s notion of ideals introduced some 75 years after Gauss’s work. For subsequent developments and generalizations the viewpoint of ideals has proven to be central to all of modern algebra. In this book we present both approaches to the class group, ﬁrst the older version just in terms of forms, then the later version using ideals. Preface. Preview. The Farey Diagram. Continued Fractions. Symmetries of the Farey Diagram. Quadratic Forms. Classification of Quadratic Forms. Representations by Quadratic Forms. The Class Group for Quadratic Forms. Quadratic Fields. Bibliography. Glossary of Nonstandard Terminology. Tables. Index

Get This Torrent

Hatcher A. Topology of Numbers_2022.pdf 19.0 MB

Details for: Hatcher A. Topology of Numbers_2022

hatcher topology numbers2022