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Dedekind and Riemann
However, several of his students became influential mathematicians, among them Richard Dedekind, Bernhard Riemann, and Friedrich Bessel.
When the Riemann hypothesis is formulated for Dedekind zeta-functions, it is known as the extended Riemann hypothesis ( ERH ) and when it is formulated for Dirichlet L-functions, it is known as the generalized Riemann hypothesis ( GRH ).
Dirichlet enjoyed his time in Göttingen as the lighter teaching load allowed him more time for research and, also, he got in close contact with the new generation of researchers, especially Richard Dedekind and Bernhard Riemann.
The idea of an analogy between number fields and Riemann surfaces goes back to Richard Dedekind and Heinrich M. Weber in the nineteenth century.
The analogy with the Riemann surface case was already pointed out by Richard Dedekind and Heinrich M. Weber in the nineteenth century.
The 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.
In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζ < sub > K </ sub >( s ), is a generalization of the Riemann zeta function — which is obtained by specializing to the case where K is the rational numbers Q.
Analogously to the Riemann zeta function, the values of the Dedekind zeta function at integers encode ( at least conjecturally ) important arithmetic data of the field K. For example, the analytic class number formula relates the residue at s = 1 to the class number h ( K ) of K, the regulator R ( K ) of K, the number w ( K ) of roots of unity in K, the absolute discriminant of K, and the number of real and complex places of K. Another example is at s = 0 where it has a zero whose order r is equal to the rank of the unit group of O < sub > K </ sub > and the leading term is given by
More generally, the Riemann zeta function and the L-series can be replaced by the Dedekind zeta function of an algebraic number field or a Hecke L-series.
His work Theorie der algebraischen Functionen einer Veränderlichen ( with Dedekind ) established an algebraic foundation for Riemann surfaces, allowing a purely algebraic formulation of the Riemann-Roch theorem.
Since G has an irreducible representation of degree 2, an Artin L-function for such a representation occurs, squared, in the factorisation of the Dedekind zeta-function for such a number field, in a product with the Riemann zeta-function ( for the trivial representation ) and an L-function of Dirichlet's type for the signature representation.

Dedekind and Cantor
That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the CantorDedekind axiom.
Cantor also mentioned the idea in his letters to Richard Dedekind ( text in square brackets not present in original ):
In his professorial doctoral dissertation, On the Concept of Number ( 1886 ) and in his Philosophy of Arithmetic ( 1891 ), Husserl sought, by employing Brentano's descriptive psychology, to define the natural numbers in a way that advanced the methods and techniques of Karl Weierstrass, Richard Dedekind, Georg Cantor, Gottlob Frege, and other contemporary mathematicians.
During his honeymoon in the Harz mountains, Cantor spent much time in mathematical discussions with Richard Dedekind, whom he had met two years earlier while on Swiss holiday.
In 1882, the mathematical correspondence between Cantor and Dedekind came to an end, apparently as a result of Dedekind's declining the chair at Halle.
At that Congress, Cantor renewed his friendship and correspondence with Dedekind.
Dedekind, whom Cantor befriended in 1872, cited this paper later that year, in the paper where he first set out his celebrated definition of real numbers by Dedekind cuts.
Owing to the gigantic simultaneous efforts of Frege, Dedekind and Cantor, the infinite was set on a throne and revelled in its total triumph.
As Cantor and Dedekind were developing more abstract versions of Stevin's continuum, Paul du Bois-Reymond wrote a series of papers on infinitesimal-enriched continua based on growth rates of functions.
In the second half of the nineteenth century, the calculus was reformulated by Augustin-Louis Cauchy, Bernard Bolzano, Karl Weierstrass, Cantor, Dedekind, and others using the ( ε, δ )- definition of limit and set theory.
* the various ( but equivalent ) constructions of the real numbers by Dedekind and Cantor resulting in the modern axiomatic definition of the real number field ;
The first three, due to Georg Cantor / Charles Méray, Richard Dedekind and Karl Weierstrass / Otto Stolz all occurred within a few years of each other.
Bolzano's posthumously published work Paradoxien des Unendlichen ( The Paradoxes of the Infinite ) was greatly admired by many of the eminent logicians who came after him, including Charles Sanders Peirce, Georg Cantor, and Richard Dedekind.
There are also papers on Dedekind, Cantor, and Russell.
Given the CantorDedekind axiom, this algorithm can be regarded as an algorithm to decide the truth of any statement in Euclidean geometry.
Mistakes and philosophical weaknesses in reasoning about infinitesimal numbers in the work of Gottfried Leibniz, Johann Bernoulli, Leonhard Euler, Augustin-Louis Cauchy, and others were the reason that they were originally abandoned for the more cumbersome real number-based arguments developed by Georg Cantor, Richard Dedekind, and Karl Weierstrass, which were perceived as being more rigorous by Weierstrass's followers.
Of a totally different orientation < nowiki > the " Old Formalist School " of Richard Dedekind | Dedekind, Georg Cantor | Cantor, Giuseppe Peano | Peano, Ernst Zermelo | Zermelo, and Louis Couturat | Couturat, etc .< nowiki ></ nowiki > was the Pre-Intuitionist School, mainly led by Poincaré, Borel and Lebesgue.
In mathematical logic, the phrase CantorDedekind axiom has been used to describe the thesis that the real numbers are order-isomorphic to the linear continuum of geometry.

Dedekind and they
Multiplication is defined for ideals, and the rings in which they have unique factorization are called Dedekind domains.
Principal ideal domains are noetherian, they are integrally closed, they are unique factorization domains and Dedekind rings.
Many more authors state theorems for Dedekind domains with the implicit proviso that they may require trivial modifications for the case of fields.
It was shown that while rings of algebraic integers do not always have unique factorization into primes ( because they need not be principal ideal domains ), they do have the property that every proper ideal admits a unique factorization as a product of prime ideals ( that is, every ring of algebraic integers is a Dedekind domain ).
In this sense, the ideal class group measures how far R is from being a principal ideal domain, and hence from satisfying unique prime factorization ( Dedekind domains are unique factorization domains if and only if they are principal ideal domains ).
Global L-functions can be associated to elliptic curves, number fields ( in which case they are called Dedekind zeta-functions ), Maass forms, and Dirichlet characters ( in which case they are called Dirichlet L-functions ).
Two fields are called arithmetically equivalent if they have the same Dedekind zeta function.

Dedekind and had
Dedekind then constructed the real numbers by Dedekind cuts, in which irrational numbers are formally defined, which serve to fill the " gaps " between rational numbers, thereby creating a complete set: the continuum of real numbers, which had already been developed by Simon Stevin in terms of decimal expansions.
By the 20th century, algebraists and number theorists had come to realize that the condition of being a PID is rather delicate, whereas the condition of being a Dedekind domain is quite robust.
A paper published by Dedekind in 1900 had lattices as its central topic: He described the free modular lattice generated by three elements, a lattice with 28 elements.

Dedekind and all
In mathematics, a Dedekind cut, named after Richard Dedekind, is a partition of the rational numbers into two non-empty parts A and B, such that all elements of A are less than all elements of B, and A contains no greatest element.
More generally, a Dedekind cut is a partition of a totally ordered set into two non-empty parts ( A and B ), such that A is closed downwards ( meaning that for all a in A, x ≤ a implies that x is in A as well ) and B is closed upwards, and A contains no greatest element.
Thus a Dedekind domain is a domain which satisfies any one, and hence all four, of ( DD1 ) through ( DD4 ).
All principal ideal domains and therefore all discrete valuation rings are Dedekind domains.
As above, this includes all the examples considered by Kummer and Dedekind and was the motivating case for the general definition, and these remain among the most studied examples.
* We form the set of real numbers as the set of all Dedekind cuts of, and define a total ordering on the real numbers as follows:
More generally, the PBW theorem as formulated above extends to cases such as where ( 1 ) L is a flat K-module, ( 2 ) L is torsion-free as an abelian group, ( 3 ) L is a direct sum of cyclic modules ( or all its localizations at prime ideals of K have this property ), or ( 4 ) K is a Dedekind domain.
The Dedekind zeta function of K has an Euler product which is a product over all the prime ideals P of O < sub > K </ sub >
For instance, in the construction of the real numbers from Dedekind cuts of rational numbers, the rational number is identified with the set of all rational numbers less than, even though the two are obviously not the same thing ( as one is a rational number and the other is a set of rational numbers ).

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