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59 relations: Abelian extension, Algebraic K-theory, Algebraic number field, American Mathematical Society, Analytic continuation, Artin conductor, Artin L-function, Cardinality, Carl Ludwig Siegel, Class number formula, Cohomology, Complex number, Complex plane, Dedekind domain, Dirichlet character, Dirichlet L-function, Dirichlet series, Dirichlet's unit theorem, Discriminant of an algebraic number field, Erich Hecke, Euler product, Functional equation (L-function), Galois extension, Galois group, Gamma function, Gassmann triple, Generalized Riemann hypothesis, Graduate Texts in Mathematics, Hasse–Weil zeta function, Ideal (ring theory), Ideal class group, Ideal norm, Index of a subgroup, Irreducible representation, Jacobi symbol, List of zeta functions, Mathematics, Meromorphic function, Motive (algebraic geometry), Motivic L-function, Peter Gustav Lejeune Dirichlet, Prime ideal, Quadratic field, Quadratic reciprocity, Quotient ring, Rank of an abelian group, Rational number, Regular representation, Residue (complex analysis), Richard Dedekind, ..., Riemann zeta function, Ring of integers, Spectrum of a ring, Springer Science+Business Media, Stephen Lichtenbaum, Totally real number field, Unit (ring theory), Vorlesungen über Zahlentheorie, Zeros and poles. Expand index (9 more) »
Abelian extension
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian.
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Algebraic K-theory
Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory.
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Algebraic number field
In mathematics, an algebraic number field (or simply number field) F is a finite degree (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.
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American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.
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Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function.
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Artin conductor
In mathematics, the Artin conductor is a number or ideal associated to a character of a Galois group of a local or global field, introduced by as an expression appearing in the functional equation of an Artin L-function.
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Artin L-function
In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced in the 1923 by Emil Artin, in connection with his research into class field theory.
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Cardinality
In mathematics, the cardinality of a set is a measure of the "number of elements of the set".
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Carl Ludwig Siegel
Carl Ludwig Siegel (December 31, 1896 – April 4, 1981) was a German mathematician specialising in number theory and celestial mechanics.
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Class number formula
In number theory, the class number formula relates many important invariants of a number field to a special value of its Dedekind zeta function.
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Cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex.
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Complex number
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
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Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis.
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Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals.
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Dirichlet character
In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z. Dirichlet characters are used to define Dirichlet ''L''-functions, which are meromorphic functions with a variety of interesting analytic properties.
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Dirichlet L-function
In mathematics, a Dirichlet L-series is a function of the form Here χ is a Dirichlet character and s a complex variable with real part greater than 1.
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Dirichlet series
In mathematics, a Dirichlet series is any series of the form where s is complex, and a_n is a complex sequence.
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Dirichlet's unit theorem
In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet.
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Discriminant of an algebraic number field
In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the (ring of integers of the) algebraic number field.
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Erich Hecke
Erich Hecke (20 September 1887 – 13 February 1947) was a German mathematician.
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Euler product
In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers.
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Functional equation (L-function)
In mathematics, the L-functions of number theory are expected to have several characteristic properties, one of which is that they satisfy certain functional equations.
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Galois extension
In mathematics, a Galois extension is an algebraic field extension E/F that is normal and separable; or equivalently, E/F is algebraic, and the field fixed by the automorphism group Aut(E/F) is precisely the base field F. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.
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Galois group
In mathematics, more specifically in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension.
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Gamma function
In mathematics, the gamma function (represented by, the capital Greek alphabet letter gamma) is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers.
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Gassmann triple
In mathematics, a Gassmann triple (or Gassmann-Sunada triple) is a group G together with two faithful actions on sets X and Y, such that X and Y are not isomorphic as G-sets but every element of G has the same number of fixed points on X and Y. They were introduced by Fritz Gassmann in 1926.
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Generalized Riemann hypothesis
The Riemann hypothesis is one of the most important conjectures in mathematics.
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Graduate Texts in Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag.
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Hasse–Weil zeta function
In mathematics, the Hasse–Weil zeta function attached to an algebraic variety V defined over an algebraic number field K is one of the two most important types of L-function.
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Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.
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Ideal class group
In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of, and is its subgroup of principal ideals.
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Ideal norm
In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension.
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Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G. For example, if H has index 2 in G, then intuitively half of the elements of G lie in H. The index of H in G is usually denoted |G: H| or or (G:H).
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Irreducible representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper subrepresentation (\rho|_W,W), W \subset V closed under the action of \. Every finite-dimensional unitary representation on a Hermitian vector space V is the direct sum of irreducible representations.
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Jacobi symbol
Jacobi symbol for various k (along top) and n (along left side).
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List of zeta functions
In mathematics, a zeta function is (usually) a function analogous to the original example: the Riemann zeta function Zeta functions include.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Meromorphic function
In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of D except for a discrete set of isolated points, which are poles of the function.
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Motive (algebraic geometry)
In algebraic geometry, a motive (or sometimes motif, following French usage) denotes 'some essential part of an algebraic variety'.
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Motivic L-function
In mathematics, motivic L-functions are a generalization of Hasse–Weil ''L''-functions to general motives over global fields.
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Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet (13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and 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 function.
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Prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers.
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Quadratic field
In algebraic number theory, a quadratic field is an algebraic number field K of degree two over Q, the rational numbers.
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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.
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Quotient ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient groups of group theory and the quotient spaces of linear algebra.
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Rank of an abelian group
In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset.
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Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.
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Regular representation
In mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself by translation.
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Residue (complex analysis)
In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities.
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Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory), axiomatic foundation for the natural numbers, algebraic number theory and the definition of the real numbers.
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Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function,, is a function of a complex variable s that analytically continues the sum of the Dirichlet series which converges when the real part of is greater than 1.
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Ring of integers
In mathematics, the ring of integers of an algebraic number field is the ring of all integral elements contained in.
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Spectrum of a ring
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by \operatorname(R), is the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space.
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Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
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Stephen Lichtenbaum
Stephen Lichtenbaum (1939 in Brooklyn) is an American mathematician who is working in the fields of algebraic geometry, algebraic number theory and algebraic K-theory.
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Totally real number field
In number theory, a number field K is called totally real if for each embedding of K into the complex numbers the image lies inside the real numbers.
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Unit (ring theory)
In mathematics, an invertible element or a unit in a (unital) ring is any element that has an inverse element in the multiplicative monoid of, i.e. an element such that The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group for this operation.
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Vorlesungen über Zahlentheorie
Vorlesungen über Zahlentheorie (German for Lectures on Number Theory) is the name of several different textbooks of number theory.
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Zeros and poles
In mathematics, a zero of a function is a value such that.
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Redirects here:
Arithmetically equivalent number fields, Dedekind zeta-function.
References
[1] https://en.wikipedia.org/wiki/Dedekind_zeta_function
