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Lectures on Number Theory

P. G. L. Dirichlet
American Mathematical Society
Publication Date: 
Number of Pages: 
History of Mathematics Source Series 16
[Reviewed by
Fernando Q. Gouvêa
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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." Editions and translations of such works have been in short supply, and those of us who are interested in the history of mathematics can often be heard to lament that this or that crucial source is no longer available. 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.

What about the current book? Dirichlet's Lectures was the first real textbook in number theory, intended for (advanced) students rather than for researchers in the area. In contrast to Gauss's famously difficult Disquisitiones Arithmeticae, this is quite accessible, and could almost be used as a textbook still today. In addition to Dirichlet's lectures, the book includes several supplements by Richard Dedekind which complement the text in various ways. The two final supplements, which were the beginning of algebraic number theory, were not included here, mostly because Cambridge University Press has recently published a translation of Dedekind's Theory of Algebraic Integers (see our earlier note). I rather regret this, since I'd have enjoyed reading those famous first steps, but the regret is swallowed up by my delight in having the Dirichlet text available. 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.

  • On the divisibility of numbers
  • On the congruence of numbers
  • On quadratic residues
  • On quadratic forms
  • Determination of the class number of binary quadratic forms
  • Some theorems from Gauss's theory of circle division
  • On the limiting value of an infinite series
  • A geometric theorem
  • Genera of quadratic forms
  • Power residues for composite moduli
  • Primes in arithmetic progressions
  • Some theorems from the theory of circle division
  • On the Pell equation
  • Convergence and continuity of some infinite series
  • Index