# Number theory

original is the number theory (also: Arithmetic) a subsection of mathematics, which generally concerns itself in particular with the characteristics of the whole numbers and with the solutions of equations in the whole numbers (Diophanti equation). Out more modernView covers it all mathematical theories, which developed historically from these questions.

## of subsections

the different subsections of the number theory generally becomeaccording to the methods differentiated, according to which pay-theoretical questions are worked on.

### elementary number theory

from the antique one into the seventeenth century maintained ground the number theory as basicconstant discipline and got along without other mathematical subsections. Their only helping meanswere the characteristics of the whole numbers, in particular prime factorization (fundamental principle of arithmetic), divisibility and counting on congruences. Such an pure approach is called also elementary number theory. Important results, which obtain themselves with the help of elementary methodsleave, are the small sentence of Fermat and its Verallgemeinerung, the sentence of Euler, the Chinese remainder set, the sentence of Wilson and the Euclidean algorithm.

### analytic number theory

as the first noticed Euler that oneMethods of the analysis and function theory to use knew, in order to solve pay-theoretical questions. One calls such an approach analytic number theory. Important problems, which were solved with analytic methods, concern usually statistic questions about the distribution of prime numbers and thatAsymptotik, like for example the prime number set of Gauss and the dirichletsche sentence over prime numbers in arithmetic progressions. Besides analytic methods served also to prove the Transzendenz of numbers like the circle number π or the Euler number of e. In the connectionwith the prime number set also the Zeta functions emerged first, which are today article both analytic and algebraic research. The probably most famous Zeta function is the Riemann Zeta function, starting point of the Riemann assumption.

### algebraic number theory and arithmeticGeometry

one of the large milestones of the number theory formed the discovery of the square reciprocity law. It showed that one can solve questions of the solubility of diophantischer equations in the whole numbers by the transition to other counting ranges more simply (square number bodies, Gauss numbers). For this one regards finite extensions of the rational numbers, so-called algebraic number bodies (from where also the name algebraic number theory comes of). Elements of number bodies are zeros of polynomials with rational coefficients. These number bodies contain the whole numbers similar subsets, those Entireness rings. They behave in many respects like the ring of the whole numbers. The clear dismantling in prime numbers applies however only in few number bodies of the class number of 1. However entireness rings are Dedekindringe and each broken ideal possess therefore oneclear dismantling into prime ideals. The analysis of these algebraic number bodies is very complicated and requires methods of almost all subsections of the abstract mathematics, in particular algebra, topology, analysis, function theory (in particular the theory of the module forms), geometry and representation theory. The algebraic number theory concerns itself further with the study of algebraic function bodies over finite bodies, whose theory runs to a large extent similarly to the theory of the number bodies. Algebraic number and function body are summarized under the name” global bodies “. Often placesit itself as fruitfully out, questions” locally “, i.e. to regard for each prime number p individually. This procedure leads in the case of the whole numbers to the p-adischen numbers, generally to local bodies.

For the formulation of the modern algebraic number theorythe language of homologischen algebra and in particular the originally topological concepts of the Kohomologie , Homotopie are essential and derived radio gates. High points of the algebraic number theory are the class body theory and the Iwasawa theory.

After the new formulation of algebraic geometry by Grothendieckand in particular after introduction of the patterns it turned out (in the second half of the twentieth century) that the number theory can be regarded as a special case of algebraic geometry. The modern algebraic number theory becomes therefore also as geometrical number theory orarithmetic geometry designates, in which the term of the pattern plays a central role.

To each number body belongs a Zeta function, whose analytic behavior reflects the arithmetic of the number body. Also for the Dedekind Zeta functions the riemannsche assumption is generally unproven.For finite bodies their statement is contained of algebraic geometry and from Pierre Deligne with means was solved in the famous because assumptions, for which it got 1978 the falling DS medal.

### algorithmic number theory

the algorithmic number theory is a branch thatNumber theory, which with the arising of computers broad interest encountered. This branch of the number theory is occupied with how pay-theoretical problems can be converted algorithmically efficiently. Important questions are, whether a large number is prime, the factorizing of large numbers andclosely the associated question about an efficient computation of the discrete logarithm. In addition there are in the meantime algorithms for the computation of class numbers, Kohomologiegruppen and the K-theory of algebraic number bodies.

## applications of the number theory

applications of the number theory are incryptography, in particular with the question about the security of the data communication in the Internet. Here both elementary methods of the number theory (prime factorization, approximately with RSA or ElGamal ) find, and advanced methods of the algebraic number theory, as for instance the codingover elliptical curves (HIT A CORNER) application spreads.

A further area of application is the coding theory, which relies in its modern form on the theory of the algebraic function bodies.

## historical development

### number theory in the antique oneand in the Middle Ages

the first written proofs of the number theory are enough to approx. 2000 v. Chr. back. The Babylonier and Egyptian knew the numbers in this time already smaller one million, the Quadratzahlen as well as some Pythagorean Tripel.

The systematic developmentthe number theory began however only in the first millenium v. Chr. in the antique Greece. Most outstanding representative is Euklid (approx. 300 v. Chr.), which introduced the method of the mathematical proof invented by Pythagoras to the number theory. Its most famous work, Euklids of elements, into the eight tenth century as standard text book for geometry and number theory one used. The volumes 7, 8 and 9 concern themselves thereby with pay-theoretical questions, among other things the definition of the prime number, a procedure for the computation of the largest commonDivisor (Euclidean algorithm) and the proof of the existence infinitely many prime numbers (sentence of Euklid).

In the year 250 v. Chr. concerned himself the Greek mathematician Diophant first with the equations of the same name, which he with linear substitutions upwell-known cases to reduce tried and actually some simple equations solved. Diophants Hauptwerk is the Arithmetica.

The Greeks raised many important arithmetic questions, which are unresolved until today partially, like e.g. the problem of the prime number twins, that perfect numbers or the triangle numbers (whereby the latter of J.B. Tunnel on the assumption of a weak form of the assumption of Birch and Swinnerton Dyer as almost solved to be regarded can) or their solution many thousands of years in requirement took and exemplary for thoseStand for development of the number theory.

With the fall of the Greek states also the bloom time of the number theory in Europe expired. From this time only the name of the Leonardo is di Pisa (approx. 1200 n. Chr.) (Fibonacci) considerably, itself besideSequences of numbers and the dissolution of equations by radicals also with diophantischen equations concerned. At the end of the Middle Ages marine Mersenne went into action, which discovered the Mersenne prime numbers.

### number theory in the early modern times

the first important representative of the number theorythe modern times was Pierre de Fermat (1607-1665). It proved the small sentence of Fermat, examined the representability of a number as sum of two squares and invented the method of the infinite descent, with which it the large set up by itSentence of Fermat in case of [itex] n=4< /math> to solve could. The general solution of the large sentence inspired the methods of the number theory over the next centuries into the modern trend.

The eight tenth century of the number theory is controlled particularly by three mathematicians: Leonhard Euler (1707-1783), of Joseph Louis lying rank (1736-1813) and Adrien Marie Legendre (1752-1833).

Euler's complete work is very extensive, and here only a small part of its pay-theoretical working can be called. It introduced the analytic methods to the number theory and foundin this way a new proof for the infinity of the quantity of the prime numbers. It invented the pay-theoretical functions, in particular the Euler Phifunktion, examined partitions and already regards one hundred years before Bernhard Riemann the riemannsche Zeta function. He discoveredthe square reciprocity law, could not prove it however, did not show the fact that the Euler number is irrational e and solves the large sentence from Fermat in the case [itex] n=3< /math>.

, Justifies the systematic theory that proves the sentence of Wilson to lying rank pellschen equation and the theory of the square forms, which will only find their conclusion in the first half of the twentieth century.

Legendre introduces the putting RH symbol to the number theory and formulates the square reciprocity law in its current form. ItsProof uses however the infinity of the quantity of the prime numbers in arithmetic progressions, which only 1832 is proven by Peter Gustav Lejeune Dirichlet.

The next large break in the history of the number theory becomes by working Carl Friedrich Gauss (1777-1855)determined. Gauss gave as first two complete proofs for the square reciprocity law. It developed Legendres further theory of the square forms and developed it to a complete theory. It created the arithmetic of the square number bodies, whereby it however into thatTo concept formations of the square forms verwurzelt remained. In this way he found the law of decomposition of the prime numbers in [itex] \ mathbb {Z} [i] [/itex], the Gauss numbers. Likewise it examined first the circling hurrying bodies, i.e. the solutions of the equation [itex] x^ {P1} = 1 [/itex] anddeveloped the calculation of the Gauss sums, which has to today great importance. He discovered in addition the Gauss prime number set, could not him however up to his death not prove. Altogether one can say that the number theory only by Gauss oneindependent and systematically arranged discipline became.

### the nineteenth century

the nineteenth century is above all the bloom time of the analytic number theory. Under Niels Henrik Abel (1802-1829), Carl Gustav Jacobi (1804-1851), Gotthold iron stone (1823-1852) and Peter Gustav LejeuneDirichlet (1805-1859) is developed the theory of the elliptical functions, which finally places the theory of the elliptical curves on a completely new foundation. Dirichlet invents the term of the L-row and proves thereby the prime number set in arithmetic progressions. Dirichlet and iron stoneuse the theory of the module forms by the number of representations of a number as sum of four and/or. to examine five squares. The unit set of Dirichlet (also in purely algebraic area out-did), is today one the Grundpfeiler thatalgebraic number theory.Bernhard Riemann (1826-1866) discovered and proved the functional equation of the Riemann Zeta function and set up profound assumptions, which brought analytic characteristics to this function with arithmetic in connection.

For entire mathematics very important, was short working of the Evariste Galois (1811-1832), that the Galoistheorie developed and thus many old questions, as the quadrature of the circle, the construction of n-corners by means of circles and ruler and the dissolving barness of polynomial equations by radicals clarified. The Galoistheorie plays today in thatNumber theory an exposed role.

In the algebraic school of the nineteenth century above all Ernst Eduard grief ( 1810-1893) is Leopold Kronecker (1823-1891), to call and smelling pool of broadcasting corporations Dedekind (1831-1916). These justified together the cornerstone of the modern structural view of algebra, in particular thoseTheory of the groups, rings and ideals, as well as the algebraic number body. Kronecker introduces the term of a divisor and discovers that today sentence sentence designated by crowning hitting a corner he weber, according to which all abelschen extensions are contained in a circling hurrying body. Grief provedthe existence of Ganzheitsbasen showed the large sentence of Fermat for all regular prime numbers and Dedekind in number bodies.

### the twentieth century and the modern trend

the twentieth century brought finally some solutions to the number theory, after those it in such a wayfor a long time had researched, i.e.:

• The complete solution of the simplest (nontrivial) type of the Diophanti equation: the square form
• with class body theory and Iwasawatheorie a by no means complete, but structurally satisfying description of the abelschen and cyclic number bodies, those to a general reciprocity law for arbitraryPower remainders led, the Arti reciprocity law.
• (Still unproven) the solution of the secondarysimplest type of the Diophanti equation: the elliptical curves

innovative for the number theory of the twentieth century the discovery of the p-adischen numbers was through short Hensel. Constructing on its workthe mathematicians Minkowski and Helmut Hasse could solve the problem of the square forms: a square form [itex] f (x, y) \ in \ mathbb {Q} [X, Y]< /math> exactly then a rational solution has [itex] (x, y) \ in \ mathbb {Q} ^2 [/itex], if it a solution in each body [itex] \ mathbb {Q} _p[/itex] possesses. This famous sentence of hate Minkowski supplies thereby a first example of a restaurant global principle, which become very important for the modern number theory.

Constructing on the work of grief, the class body theory at the beginning of the twentieth century of one becomeswhole row of mathematicians develops. Among them above all David Hilbert, Helmut Hasse, Philipp Furtwängler is, to call Teiji Takagi and Emil Artin whereby Takagi proved the important existence set, from which Artin its famous reciprocity law derives. Onecomplete computation of the Hilbertsymbols and thus the practical application of the reciprocity law, however only the mathematician gave to Helmut Brückner in the second half of the twentieth century. Into the modern language of the Gruppenkohomologie, abstract harmonious analysis and representation theory the class body theory becamefrom mathematicians such as John and smelling pool of broadcasting corporations of long country would do translated. Long country assumed large Verallgemeinerungen of the class body theory and put so the foundation-stone for the long land program, which is an important part of the active pay-theoretical research.

For zyklotomische bodies Kenkichi Iwasawa finally developed the Iwasawatheorie, which could explain these bodies still better. With these bodies certain p-adische L-rows are linked. The Haupvermutung of the Iwasawatheorie, which explains the different possibilities these L-rows to define for equivalent, became for total-real number bodies of Berry Mazur and Andrew Wiles with the end of the eighties proved.

Also within the range of the elliptical curves the number theoreticians made large progress. Louis Joel Mordell examined the group law of elliptical curves and showed that the group of the rational points produces always finallyis, a simple version of the sentence of Modell-Weil. Carl Ludwig seal could finally show that each elliptical curve possesses only finally many whole solutions (sentence of seal). Thus was the problem of the whole and rational points on ellipticalCurves become open to attack.

Mordell assumed that for curves of the sex > 1 (which no more elliptical curves are) the quantity of the rational points always is finite (Mordell assumption). This proved the German mathematician Gerd Faltings, for which he 1986 the falling DS medal got. Thus it was shown that the large sentence of Fermat could have at the most finally many solutions.

The work of B. meant a large break-through. Birch and H.P.F. Swinnerton Dyer in the second half of the twentieth century. They assumed thatan elliptical curve exactly then many rational solutions possesses infinitely, if its L-row at the point [itex] s=1< /math> a value not equal zero takes. This is a very weak form of the so-called Birch and Swinnerton Dyer assumption. Although it is unproven in principle, givesit strong theoretical and nummerische arguments for their correctness. In recent time Don Zagier and Benedikt H proved. Largely their validity for a multiplicity of elliptical curves.

To remain not unmentioned is the proof of the modularity set, by Christophe Breuil, Brian Conrad, Fred dia. moon and smelling pool of broadcasting corporations Taylor in the year 2001, after Andrew Wiles had before already proven it for most elliptical curves (1995). From (of Wiles proven) the part of the modularity set it follows in particular that the large sentencefrom Fermat is true.

Todo: Supplement and correct… (e.g. André is missing because)…

## important number theoreticians

this collection should still are extended.

## literature

• G. H. Hardy, E. M. WRIGHT: On introduction ton the theory OF numbers. Oxford University press, ISBN 0198531710
• John H. Conway, smelling pool of broadcasting corporations K. Guy: The Book OF Numbers. ISBN 0-387-97993-X
• Jürgen Neukirch: Algebraic number theory. Springer publishing house,Berlin Heidelberg New York.ISBN 3-540-542-736
• J. Neukirch, A. Schmidt, K. Wingberg: Cohomology OF NUMBER fields. Springer publishing house, Berlin Heidelberg New York, ISBN 3-540-666-710
• Jörg brothers: Introduction to the analytic number theory. Springer publishing house, Berlin ISBN 3-540-588-213
• Arnold Scholz, Bruno beautiful mountain: Introduction to the number theory. Walter de Gruyter & Ko., Berlin ISBN 3-11-004423-4
• Friedhelm PAD mountain: Elementary number theory. Spectrum academic publishing house, Berlin Heidelberg. ISBN 3-86025-453-7