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

5.0 out of 5 stars 3 customer reviews
ISBN-13: 978-0821820179
ISBN-10: 0821820176
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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
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Product Details

  • Series: History of Mathematics Source Series, V. 16 (Book 16)
  • Paperback: 275 pages
  • Publisher: American Mathematical Society (August 4, 1999)
  • Language: English
  • ISBN-10: 0821820176
  • ISBN-13: 978-0821820179
  • Product Dimensions: 0.8 x 7.2 x 10.2 inches
  • Shipping Weight: 1.2 pounds (View shipping rates and policies)
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (3 customer reviews)
  • Amazon Best Sellers Rank: #2,211,701 in Books (See Top 100 in Books)

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Format: 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.Read more ›
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Incredible. It is beyond belief that I am holding this book in my hands. It is always astounding to be reading the writings of one of the great mathematicians. This is a beautiful book, superbly translated. If you have an interest in number theory, don't hesitate, this is a classic.
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Format: Paperback
This is a translation of Dirichlet's "Vorlesungen über Zahlentheorie", edited with supplements by Dedekind. The emphasis of the book is on the class number of binary quadratic forms, that is the number of equivalence classes of binary quadratic forms with a given determinant. This is developed from scratch. The supplements are not mere technical appendices but contain results that are interesting and important by themselves, like the calculation of Gauss sums modulo composite numbers in Supplement I, the notion of genera of quadratic forms in Supplement IV, Dirichlet L-functions in Supplement VI, and cyclotomic polynomials in Supplement VII. A useful project for a historically minded mathematician would be to write a reader's guide for this work in fully modern language, as it is useful to give mathematicians a sense of what happens in old mathematics books without having to read the books cover to cover, especially to choose parts of old works to read in detail.
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