- Vorlesungen über Zahlentheorie
_de. "
Vorlesungen über Zahlentheorie " (German for "Lectures on Number Theory") is a textbook ofnumber theory written by German mathematicians P.G.L. Dirichlet andRichard Dedekind , and published in1863 .Based on Dirichlet's number theory course at the University of
Göttingen , the _de. "Vorlesungen" were edited by Dedekind and published after Dirichlet's death. Dedekind added several appendices to the _de. "Vorlesungen", in which he collected further results of Dirichlet's and also developed his own original mathematical ideas.cope
The _de. "Vorlesungen" cover topics in elementary number theory,
algebraic number theory andanalytic number theory , includingmodular arithmetic , quadratic congruences,quadratic reciprocity and binaryquadratic form s.Contents
The contents of Professor John Stillwell's 1999 translation of the _de. "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' _la. "
Disquisitiones Arithmeticae ", and Dedekind added footnotes which specifically cross-reference the relevant sections of the _la. "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 field s. 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 _de. "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 andHilbert . Dirichlet does not explicitly recognise the concept of the group that is central tomodern algebra , but many of his proofs show an implicit understanding of group theoryThe _de. "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
Note that the
Göttinger Digitalisierungszentrum has a [http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN30976923X scanned copy] of the original, 2nd edition text (in German).
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