Vorlesungen über Zahlentheorie (Lectures on Number Theory) is a textbook of number theory written by German mathematicians P.G.L. Dirichlet and Richard Dedekind, and published in 1863.

Based on Dirichlet's number theory course at the University of Göttingen, the Vorlesungen were edited by Dedekind and published after Dirichlet's death. Dedekind added several appendices to the Vorlesungen, in which he collected further results of Dirichlet's and also developed his own original mathematical ideas.

Scope

The Vorlesungen cover topics in elementary number theory, algebraic number theory and analytic number theory, including modular arithmetic, quadratic congruences, quadratic reciprocity and binary quadratic forms.

Contents

The contents of Professor John Stillwell's 1999 translation of the Vorlesungen are as follows

Chapter 1. On the divisibility of numbers

Chapter 2. On the congruence of numbers

Chapter 3. On quadratic residues

Chapter 4. On quadratic forms

Chapter 5. 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

This translation does not include Dedekind's Supplements X and XI in which he begins to develop the theory of ideals.

Chapters 1 to 4 cover similar ground to Gauss' Disquisitiones Arithmeticae, and Dedekind added footnotes which specifically cross-reference the relevant sections of the Disquisitiones. These chapters can be thought of as a summary of existing knowledge, although Dirichlet simplifies Gauss' presentation, and introduces his own proofs in some places.

Chapter 5 contains Dirichlet's derivation of the class number formula for real and imaginary quadratic fields. Although other mathematicians had conjectured similar formulae, Dirichlet gave the first rigorous proof.

Supplement VI contains Dirichlet's proof that an arithmetic progression of the form a+nd where a and d are coprime contains an infinite number of primes.

Importance

The Vorlesungen can be seen as a watershed between the classical number theory of Fermat, Jacobi and Gauss, and the modern number theory of Dedekind, Riemann and Hilbert. Dirichlet does not explicitly recognise the concept of the group that is central to modern algebra, but many of his proofs show an implicit understanding of group theory

The Vorlesungen contains two key results in number theory which were first proved by Dirichlet. The first of these is the class number formulae for binary quadratic forms. The second is a proof that arithmetic progressions contains an infinite number of primes (known as Dirichlet's theorem); this proof introduces Dirichlet L-series. These results are important milestones in the development of analytic number theory.

References

• P.G.L. Dirichlet, R. Dedekind tr. John Stillwell: Lectures on Number Theory, American Mathematical Society, 1999 ISBN 0821820176