Vorlesungen über Zahlentheorie
Encyclopedia
is a textbook of number theory
written by German
mathematicians Lejeune Dirichlet
and Richard Dedekind
, and published in 1863.
Based on Dirichlet's number theory course at the University of Göttingen, the were edited by Dedekind and published after Dirichlet's death. Dedekind added several appendices to the , in which he collected further results of Dirichlet's and also developed his own original mathematical ideas.
and analytic number theory
, including modular arithmetic
, quadratic congruences, quadratic reciprocity
and binary quadratic form
s.
's 1999 translation of the are as follows
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:
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' 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.
, 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 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.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
written by German
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
mathematicians Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet was a German mathematician with deep contributions to number theory , as well as to the theory of Fourier series and other topics in mathematical analysis; he is credited with being one of the first mathematicians to give the modern formal definition of a...
and Richard Dedekind
Richard Dedekind
Julius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.-Life:...
, and published in 1863.
Based on Dirichlet's number theory course at the University of Göttingen, the were edited by Dedekind and published after Dirichlet's death. Dedekind added several appendices to the , in which he collected further results of Dirichlet's and also developed his own original mathematical ideas.
Scope
The cover topics in elementary number theory, algebraic number theoryAlgebraic number theory
Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
and analytic number theory
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 Dirichlet's introduction of Dirichlet L-functions to give the first proof of Dirichlet's theorem on arithmetic...
, including modular arithmetic
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
, quadratic congruences, quadratic reciprocity
Quadratic reciprocity
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic which gives conditions for the solvability of quadratic equations modulo prime numbers...
and binary quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
s.
Contents
The contents of Professor John StillwellJohn Stillwell
John Stillwell is an Australian mathematician on the faculties of the University of San Francisco and Monash University.He was born in Melbourne, Australia and lived there until he went to the Massachusetts Institute of Technology for his doctorate. He received his PhD from MIT in 1970, working...
'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
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
.
The German titles of supplements X and XI are:
- Supplement X: Über die Composition der binären quadratische Formen,
- Supplement XI: Über die Theorie der ganzen algebraischen Zahlen.
Chapters 1 to 4 cover similar ground to Gauss'
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
, 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' 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
Quadratic field
In algebraic number theory, a quadratic field is an algebraic number field K of degree two over Q. It is easy to show that the map d ↦ Q is a bijection from the set of all square-free integers d ≠ 0, 1 to the set of all quadratic fields...
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 FermatPierre de Fermat
Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...
, Jacobi
Carl Gustav Jakob Jacobi
Carl Gustav Jacob Jacobi was a German mathematician, widely considered to be the most inspiring teacher of his time and is considered one of the greatest mathematicians of his generation.-Biography:...
and Gauss
Carl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
, and the modern number theory of Dedekind, Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
and Hilbert
David Hilbert
David Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
. Dirichlet does not explicitly recognise the concept of the group
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
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
Dirichlet's theorem on arithmetic progressions
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers a and d, there are infinitely many primes of the form a + nd, where n ≥ 0. In other words, there are infinitely many primes which are...
); this proof introduces Dirichlet L-series. These results are important milestones in the development of analytic number theory.