Lectures on Number Theory (History of Mathematics Source Series, V. 16) [Paperback]

Peter Gustav Lejeune Dirichlet (Author), Richard Dedekind (Author), P. G. L. Dirichlet (Author)

Book Description

ISBN-10: 0821820176 | ISBN-13: 978-0821820179 | Publication Date: August 4, 1999

This volume is a translation of Dirichlet's Vorlesungen über Zahlentheorie which includes nine supplements by Dedekind and an introduction by John Stillwell, who translated the volume. <P>Lectures on Number Theory is the first of its kind on the subject matter. It covers most of the topics that are standard in a modern first course on number theory, but also includes Dirichlet's famous results on class numbers and primes in arithmetic progressions. <P>The book is suitable as a textbook, yet it also offers a fascinating historical perspective that links Gauss with modern number theory. The legendary story is told how Dirichlet kept a copy of Gauss's Disquisitiones Arithmeticae with him at all times and how Dirichlet strove to clarify and simplify Gauss's results. Dedekind's footnotes document what material Dirichlet took from Gauss, allowing insight into how Dirichlet transformed the ideas into essentially modern form. <P>Also shown is how Gauss built on a long tradition in number theory--going back to Diophantus--and how it set the agenda for Dirichlet's work. This important book combines historical perspective with transcendent mathematical insight. The material is still fresh and presented in a very readable fashion. <P>This volume is one of an informal sequence of works within the History of Mathematics series. Volumes in this subset, "Sources", are classical mathematical works that served as cornerstones for modern mathematical thought. (For another historical translation by Professor Stillwell, see Sources of Hyperbolic Geometry, Volume 10 in the History of Mathematics series.)

Editorial Reviews

Review

"A new edition of Dirichlet's Lectures on Number Theory would be big news any day, but it's particularly gratifying to see the book appear as "the first of an informal sequence" which is to include "classical mathematical works that served as cornerstones for modern mathematical thought." So all power to the American Mathematical Society and the London Mathematical Society in their joint-venture History of Mathematics series: may the "Sources" subseries live long and prosper. [T]his is quite accessible, and could almost be used as a textbook still today. For those who like to heed Abel's admonition to "read the masters, not their students," here's a great opportunity to learn more about Number Theory." ---- MAA Online

"This is a nice English edition of Dirichlet's famous Vorlesungen über Zahlentheorie, including the nine Supplements by Dedekind, translated by John Stillwell. As one of the most important number-theoretical books of the 19th century this book needs no further description, and can be recommended to those who have problems with the German language, or to those who cannot find the German original in the library. This book should certainly have a permanent place on every mathematical bookshelf." ---- European Mathematical Society Newsletter

Language Notes

Text: English (translation)
Original Language: German

Product Details

• Paperback: 275 pages
• Publisher: American Mathematical Society (August 4, 1999)
• Language: English
• ISBN-10: 0821820176
• ISBN-13: 978-0821820179
• Product Dimensions: 9.9 x 7 x 0.6 inches
• Shipping Weight: 1.2 pounds (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (1 customer review)
• Amazon Best Sellers Rank: #1,881,950 in Books (See Top 100 in Books)

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3 of 4 people found the following review helpful:

5.0 out of 5 stars Gauss and then some, February 9, 2006

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Viktor Blasjo - See all my reviews
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This review is from: Lectures on Number Theory (History of Mathematics Source Series, V. 16) (Paperback)

Dirichlet is all about quadratic forms. But first there are three preliminary chapters on the tools we will need: unique factorisation, modulo arithmetic, quadratic reciprocity. Then in chapter 4 we get to the quadratic forms, ax^2+2bxy+cy^2. "The whole theory originates in the problem of deciding whether a given number is representable by a given form" (p. 92). (Remember, for example, that Fermat solved the case a=1, b=0, c=1 -- which integers are sums of two squares?) "The number b^2-ac, on which the properties of the form mainly depend, is called the determinant of the form", and two forms are equivalent (represent the same numbers) when one results from the other by applying a variable transformation matrix of determinant 1. And now the problem above reduces to "the two main problems in the theory of equivalence: I. To decide whether two given forms of the same determinant are equivalent. II. To find all substitutions that send one of two equivalent form to the other." (p. 100). We spend the rest of the chapter solving there two problems for any determinant D, and we work out the applications in the cases D=-1,-2,-3,-5 (which includes the theorem of Fermat above). In the cases D=-3,-5 representations are not generally unique (which we secretly think of as the manifestation of the loss of unique factorisation in Z[sqrt(-3)] and Z[sqrt(-5)]) and this goes hand in hand with the fact that the number of equivalence classes (the "class number") of forms in those cases is 2, not 1. Such matters are the motivation for Dirichlet's great contribution: the determination of the class number for any D (chapter 5). Apart from this motivation of measuring "how far quadratic integers deviate from unique prime factorisation", as Stillwell puts it (p. xvii), Dirichlet also assigns his solution of the class number problem great intrinsic beauty: "This problem is the last and most important solved in this book, and is connected with the most beautiful algebraic and analytic investigations of this century" (p. 100).

This is a pleasant book. Dirichlet is a celebrated expositor and quite rightly so. There is also an excellent 10 page introduction by Stillwell and some 70 pages of supplements by Dedekind. The most interesting supplement is certainly Dirichlet's famous L-series proof that there are infinitely many primes in essentially any arithmetic progression.