Lectures on Number Theory
About this Title
P. G. L. Dirichlet and R. Dedekind
Publication: History of Mathematics
Publication Year: 1999; Volume 16
ISBNs: 978-0-8218-2017-9 (print); 978-1-4704-3884-5 (online)
MathSciNet review: MR1710911
MSC: Primary 01A75; Secondary 01A55, 11-01
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.
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.
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.
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.
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.)
Graduate students and research mathematicians interested in number theory; mathematical historians.
Table of Contents
- 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
- Supplement I. Some theorems from Gauss’s theory of circle division
- Supplement II. On the limiting value of an infinite series
- Supplement III. A geometric theorem
- Supplement IV. Genera of quadratic forms
- Supplement V. Power residues for composite moduli
- Supplement VI. Primes in arithmetic progressions
- Supplement VII. Some theorems from the theory of circle division
- Supplement VIII. On the Pell equation
- Supplement IX. Convergence and continuity of some infinite series