Vorlesungen über Zahlentheorie
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(; German for ''Lectures on Number Theory'') is the name of several different textbooks of
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
. The best known was written by
Peter Gustav Lejeune Dirichlet Johann Peter Gustav Lejeune Dirichlet (; ; 13 February 1805 – 5 May 1859) was a German mathematician. In number theory, he proved special cases of Fermat's last theorem and created analytic number theory. In analysis, he advanced the theory o ...
and Richard Dedekind, and published in 1863. Others were written by
Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, abstract algebra and logic, and criticized Georg Cantor's work on set theory. Heinrich Weber quoted Kronecker as having said, ...
, Edmund Landau, and Helmut Hasse. They all cover elementary number theory, Dirichlet's theorem, quadratic fields and forms, and sometimes more advanced topics.


Dirichlet and Dedekind's book

Based on Dirichlet's number theory course at the
University of Göttingen The University of Göttingen, officially the Georg August University of Göttingen (, commonly referred to as Georgia Augusta), is a Public university, public research university in the city of Göttingen, Lower Saxony, Germany. Founded in 1734 ...
, the were edited by Dedekind and published after Lejeune Dirichlet's death. Dedekind added several appendices to the , in which he collected further results of Lejeune Dirichlet's and also developed his own original mathematical ideas.


Scope

The cover topics in elementary number theory,
algebraic number theory Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
and
analytic number theory In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Dir ...
, including
modular arithmetic In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to mo ...
, quadratic congruences,
quadratic reciprocity In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard st ...
and binary
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...
s.


Contents

The contents of Professor John Stillwell's 1999 translation of the 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. The German titles of supplements X and XI are: :Supplement X: Über die Composition der binären quadratische Formen (On the composition of binary quadratic forms) :Supplement XI: Über die Theorie der ganzen algebraischen Zahlen (On the theory of algebraic integers) Chapters 1 to 4 cover similar ground to Gauss' , and Dedekind added footnotes which specifically cross-reference the relevant sections of the . These chapters can be thought of as a summary of existing knowledge, although Dirichlet simplifies Gauss's 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 In algebraic number theory, a quadratic field is an algebraic number field of Degree of a field extension, degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free ...
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 can be seen as a watershed between the classical number theory of
Fermat Pierre de Fermat (; ; 17 August 1601 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his d ...
, Jacobi and
Gauss Johann Carl Friedrich Gauss (; ; ; 30 April 177723 February 1855) was a German mathematician, astronomer, Geodesy, geodesist, and physicist, who contributed to many fields in mathematics and science. He was director of the Göttingen Observat ...
, and the modern number theory of Dedekind, Riemann and
Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosophy of mathematics, philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad ...
. 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 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.


Kronecker's book

Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, abstract algebra and logic, and criticized Georg Cantor's work on set theory. Heinrich Weber quoted Kronecker as having said, ...
's book was first published in 1901 in 2 parts and reprinted by Springer in 1978. It covers elementary and algebraic number theory, including Dirichlet's theorem.


Landau's book

Edmund Landau's book ''Vorlesungen über Zahlentheorie'' was first published as a 3-volume set in 1927. The first half of volume 1 was published as ''Vorlesungen über Zahlentheorie. Aus der elementare Zahlentheorie'' in 1950, with an English translation in 1958 under the title ''Elementary number theory''. In 1969 Chelsea republished the second half of volume 1 together with volumes 2 and 3 as a single volume. Volume 1 on elementary and
additive number theory Additive number theory is the subfield of number theory concerning the study of subsets of integers and their behavior under addition. More abstractly, the field of additive number theory includes the study of abelian groups and commutative semigro ...
includes the topics such as Dirichlet's theorem, Brun's sieve, binary quadratic forms, Goldbach's conjecture, Waring's problem, and the Hardy–Littlewood work on the singular series. Volume 2 covers topics in analytic number theory, such as estimates for the error in the prime number theorem, and topics in geometric number theory such as estimating numbers of lattice points. Volume 3 covers algebraic number theory, including ideal theory, quadratic number fields, and applications to Fermat's last theorem. Many of the results described by Landau were state of the art at the time but have since been superseded by stronger results.


Hasse's book

Helmut Hasse's book ''Vorlesungen über Zahlentheorie'' was published in 1950, and is different from and more elementary than his book ''Zahlentheorie''. It covers elementary number theory, Dirichlet's theorem, and quadratic fields.


References

* P. G. Lejeune Dirichlet, R. Dedekind tr. John Stillwell: ''Lectures on Number Theory'', American Mathematical Society, 1999 The Göttinger Digitalisierungszentrum has
scanned copy
of the original, 2nd edition text (in German) published in 1871 containing supplements I–X. Supplement XI can be found in volume three of Dedekind's complete works also at th
Göttinger Digitalisierungszentrum
The 4th edition from 1894 which contains all of the supplements including Dedekind's XI is available a
Internet Archive
* * * * {{DEFAULTSORT:Vorlesungen Uber Zahlentheorie Number theory 1863 non-fiction books Mathematics books Set index articles on mathematics