Number theory
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Field theory
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History of number theory
Algebraic number theory
Algebraic number theory is a major branch of number theory that studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O...
Algebraic number theory - Wikipedia
Local field
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Dirichlet's unit theorem
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Artin reciprocity law
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Class number formula
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Theorems in algebraic number theory
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Algebraic number
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Class field theory
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Cyclotomic field
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Finite field
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Zeta and L-functions
Hilbert's ninth problem - Slideshow
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it....
Dirichlet's unit theorem
In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring OK of algebraic in...
Dirichlet's unit theorem - Wikipedia
Artin reciprocity law
The Artin reciprocity law, established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theorem in number theory that forms a central part of global class field theory. The term "...
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
We start with the following data:Then:This is the mo...
Theorems in algebraic number theory
Algebraic number
In mathematics, an algebraic number is a number that is a root of a finite, non-zero polynomial in one variable with rational coefficients (or equivalently — by clearing denominators — with integer co...
Algebraic number - Wikipedia
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields and function fields of curves over finite fields and arithmetic propert...
Cyclotomic field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational numbers. The n-th cyclotomic field Q(ζn) (where n > 2) is ...
Finite field
In algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains a finite number of elements, called its order (the size of the underlying set). As with any ...
Zeta and L-functions
Hilbert's ninth problem
Hilbert's ninth problem, from the list of 23 Hilbert's problems (1900), asked to find the most general reciprocity law for the norm residues of k-th order in a general algebraic number field, where k ...
P-adic Hodge theory
In mathematics, p-adic Hodge theory is a theory that provides a way to classify and study p-adic Galois representations of characteristic 0 local fields with residual characteristic p (such as Qp). Th...
Compatible system of ℓ-adic representations
Maillet's determinant
In mathematics, Maillet's determinant Dp is the determinant of the matrix introduced by Maillet (1913) whose entries are R(s/r) for s,r = 1, 2, ..., (p – 1)/2 ∈ Z/pZ for an odd prime ...
Hilbert's twelfth problem
Kronecker's Jugendtraum or Hilbert's twelfth problem, of the 23 mathematical Hilbert problems, is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any bas...
Extension and contraction of ideals
S-unit
In mathematics, in the field of algebraic number theory, an S-unit generalises the idea of unit of the ring of integers of the field. Many of the results which hold for units are also valid for S-uni...
S-unit - Wikipedia
Ramification
In mathematics, ramification is a geometric term used for 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign. It is also...
Bauerian extension
Lafforgue's theorem
In mathematics, Lafforgue's theorem, due to Laurent Lafforgue, completes the Langlands program for general linear groups over algebraic function fields, by giving a correspondence between automorphic ...
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. There are a number of eq...
Quadratic reciprocity - Wikipedia
Local Tate duality
In Galois cohomology, local Tate duality (or simply local duality) is a duality for Galois modules for the absolute Galois group of a non-archimedean local field. It is named after John Tate who first...
Adele ring
In algebraic number theory and topological algebra, the adele ring (other names are the adelic ring, the ring of adeles) is a self-dual topological ring built on the field of rational numbers (or, mor...
Leopoldt's conjecture
In algebraic number theory, Leopoldt's conjecture, introduced by H.-W. Leopoldt (1962, 1975), states that the p-adic regulator of a number field does not vanish. The p-adic regulator is an a...
Scholz's reciprocity law
In mathematics, Scholz's reciprocity law is a reciprocity law for quadratic residue symbols of real quadratic number fields discovered by Theodor Schönemann (1839) and rediscovered by Arnold&...
Minkowski space (number field)
Heegner point
In mathematics, a Heegner point is a point on a modular curve that is the image of a quadratic imaginary point of the upper half-plane. They were defined by Bryan Birch and named after Kurt Heegner, w...
Supersingular prime (for an elliptic curve)
In algebraic number theory, a supersingular prime is a prime number with a certain relationship to a given elliptic curve. If the curve E defined over the rational numbers, then a prime p is supersin...
Vector space
A vector space is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars in this context. Scalars are often taken to be real numbers, ...
Vector space - Wikipedia