Logo
Unionpedia
Communication
Get it on Google Play
New! Download Unionpedia on your Android™ device!
Free
Faster access than browser!
 

Algebraic number field

Index 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. [1]

176 relations: Abelian extension, Absolute Galois group, Absolute value, Absolute value (algebra), Abstract algebra, Addition, Adele ring, Adelic algebraic group, Affine variety, Algebraic closure, Algebraic curve, Algebraic extension, Algebraic geometry, Algebraic integer, Algebraic K-theory, Algebraic number, Algebraic number field, Algebraic number theory, American Mathematical Society, Analytic continuation, André Weil, Arithmetic progression, Artin reciprocity law, Bilinear form, Branched covering, Brauer group, Carl Friedrich Gauss, Cauchy sequence, Cayley–Hamilton theorem, Characteristic polynomial, Chebotarev's density theorem, Class field theory, Class number formula, Class number problem, Commutator subgroup, Completion (algebra), Complex conjugate, Complex number, Computer algebra system, Coprime integers, Countable set, Cyclotomic field, Dedekind domain, Dedekind zeta function, Degree of a field extension, Determinant, Dimension (vector space), Dirichlet L-function, Dirichlet's theorem on arithmetic progressions, Dirichlet's unit theorem, ..., Discrete valuation, Discrete valuation ring, Discriminant, Discriminant of an algebraic number field, Division algebra, Duality (mathematics), Eisenstein integer, Euclidean domain, Euclidean vector, Euler's totient function, Fiber (mathematics), Field (mathematics), Field extension, Field norm, Field of fractions, Field trace, Finite field, Finitely generated module, Frobenius endomorphism, Function field of an algebraic variety, Functional equation, Fundamental theorem of Galois theory, Galois cohomology, Galois group, Gaussian integer, Gaussian rational, Genus field, Geometry, Geometry of numbers, Global field, Glossary of algebraic geometry, Group action, Group cohomology, Hasse principle, Hilbert class field, Homogeneous function, Ideal (ring theory), Ideal class group, Image (mathematics), Imaginary unit, Integer matrix, Integral domain, Integrally closed, Intermediate value theorem, Invariant (mathematics), Inverse element, Iwasawa theory, Kronecker–Weber theorem, Krull dimension, Kummer theory, Kurt Hensel, L-function, Limit of a sequence, Linear combination, Linear function, List of number fields with class number one, Local analysis, Local class field theory, Local field, Local ring, Localization of a ring, Mathematics, Matrix (mathematics), Matrix multiplication, Maximal ideal, Minkowski's theorem, Modular arithmetic, Monic polynomial, Monogenic field, Multiplication, Noetherian ring, Ordered pair, Ostrowski's theorem, P-adic analysis, P-adic number, Polynomial, Prime element, Prime ideal, Primitive element theorem, Quadratic field, Quadratic integer, Ramification (mathematics), Ramification group, Rational number, Ray class field, Real number, Regular representation, Residue (complex analysis), Residue field, Resultant, Richard Dedekind, Riemann surface, Riemann zeta function, Ring (mathematics), Ring of integers, Root of unity, S-unit, Scalar multiplication, Scheme (mathematics), Serge Lang, Set (mathematics), Sheaf (mathematics), Simple extension, Special values of L-functions, Spectrum of a ring, Splitting of prime ideals in Galois extensions, Springer Science+Business Media, Square-free element, Subring, Symmetric matrix, Tate duality, Tensor product of fields, Topological ring, Topology, Totally imaginary number field, Totally real number field, Trace (linear algebra), Tuple, Ultrametric space, Uncountable set, Unique factorization domain, Unit (ring theory), Vector space, Weil's conjecture on Tamagawa numbers, Zero divisor, Zeros and poles. Expand index (126 more) »

Abelian extension

In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian.

New!!: Algebraic number field and Abelian extension · See more »

Absolute Galois group

In mathematics, the absolute Galois group GK of a field K is the Galois group of Ksep over K, where Ksep is a separable closure of K. Alternatively it is the group of all automorphisms of the algebraic closure of K that fix K. The absolute Galois group is well-defined up to inner automorphism.

New!!: Algebraic number field and Absolute Galois group · See more »

Absolute value

In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.

New!!: Algebraic number field and Absolute value · See more »

Absolute value (algebra)

In mathematics, an absolute value is a function which measures the "size" of elements in a field or integral domain.

New!!: Algebraic number field and Absolute value (algebra) · See more »

Abstract algebra

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

New!!: Algebraic number field and Abstract algebra · See more »

Addition

Addition (often signified by the plus symbol "+") is one of the four basic operations of arithmetic; the others are subtraction, multiplication and division.

New!!: Algebraic number field and Addition · See more »

Adele ring

In mathematics, the adele ring (also adelic ring or ring of adeles) is defined in class field theory, a branch of algebraic number theory.

New!!: Algebraic number field and Adele ring · See more »

Adelic algebraic group

In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group G over a number field K, and the adele ring A.

New!!: Algebraic number field and Adelic algebraic group · See more »

Affine variety

In algebraic geometry, an affine variety over an algebraically closed field k is the zero-locus in the affine ''n''-space k^n of some finite family of polynomials of n variables with coefficients in k that generate a prime ideal.

New!!: Algebraic number field and Affine variety · See more »

Algebraic closure

In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.

New!!: Algebraic number field and Algebraic closure · See more »

Algebraic curve

In mathematics, a plane real algebraic curve is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables.

New!!: Algebraic number field and Algebraic curve · See more »

Algebraic extension

In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e. which contain transcendental elements, are called transcendental.

New!!: Algebraic number field and Algebraic extension · See more »

Algebraic geometry

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.

New!!: Algebraic number field and Algebraic geometry · See more »

Algebraic integer

In algebraic number theory, an algebraic integer is a complex number that is a root of some monic polynomial (a polynomial whose leading coefficient is 1) with coefficients in (the set of integers).

New!!: Algebraic number field and Algebraic integer · See more »

Algebraic K-theory

Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory.

New!!: Algebraic number field and Algebraic K-theory · See more »

Algebraic number

An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently – by clearing denominators – with integer coefficients).

New!!: Algebraic number field and Algebraic number · See more »

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.

New!!: Algebraic number field and Algebraic number field · See more »

Algebraic number theory

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations.

New!!: Algebraic number field and Algebraic number theory · See more »

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.

New!!: Algebraic number field and American Mathematical Society · See more »

Analytic continuation

In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function.

New!!: Algebraic number field and Analytic continuation · See more »

André Weil

André Weil (6 May 1906 – 6 August 1998) was an influential French mathematician of the 20th century, known for his foundational work in number theory, algebraic geometry.

New!!: Algebraic number field and André Weil · See more »

Arithmetic progression

In mathematics, an arithmetic progression (AP) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.

New!!: Algebraic number field and Arithmetic progression · See more »

Artin reciprocity law

The Artin reciprocity law, which was 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.

New!!: Algebraic number field and Artin reciprocity law · See more »

Bilinear form

In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map, where K is the field of scalars.

New!!: Algebraic number field and Bilinear form · See more »

Branched covering

In mathematics, a branched covering is a map that is almost a covering map, except on a small set.

New!!: Algebraic number field and Branched covering · See more »

Brauer group

In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras over K, with addition given by the tensor product of algebras.

New!!: Algebraic number field and Brauer group · See more »

Carl Friedrich Gauss

Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.

New!!: Algebraic number field and Carl Friedrich Gauss · See more »

Cauchy sequence

In mathematics, a Cauchy sequence, named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses.

New!!: Algebraic number field and Cauchy sequence · See more »

Cayley–Hamilton theorem

In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex field) satisfies its own characteristic equation.

New!!: Algebraic number field and Cayley–Hamilton theorem · See more »

Characteristic polynomial

In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots.

New!!: Algebraic number field and Characteristic polynomial · See more »

Chebotarev's density theorem

Chebotarev's density theorem in algebraic number theory describes statistically the splitting of primes in a given Galois extension K of the field \mathbb of rational numbers.

New!!: Algebraic number field and Chebotarev's density theorem · See more »

Class field theory

In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of local fields (one-dimensional local fields) and "global fields" (one-dimensional global fields) such as number fields and function fields of curves over finite fields in terms of abelian topological groups associated to the fields.

New!!: Algebraic number field and Class field theory · See more »

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.

New!!: Algebraic number field and Class number formula · See more »

Class number problem

In mathematics, the Gauss class number problem (for imaginary quadratic fields), as usually understood, is to provide for each n ≥ 1 a complete list of imaginary quadratic fields \mathbb(\sqrt) (for negative integers d) having class number n. It is named after Carl Friedrich Gauss.

New!!: Algebraic number field and Class number problem · See more »

Commutator subgroup

In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.

New!!: Algebraic number field and Commutator subgroup · See more »

Completion (algebra)

In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules.

New!!: Algebraic number field and Completion (algebra) · See more »

Complex conjugate

In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.

New!!: Algebraic number field and Complex conjugate · See more »

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.

New!!: Algebraic number field and Complex number · See more »

Computer algebra system

A computer algebra system (CAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists.

New!!: Algebraic number field and Computer algebra system · See more »

Coprime integers

In number theory, two integers and are said to be relatively prime, mutually prime, or coprime (also written co-prime) if the only positive integer (factor) that divides both of them is 1.

New!!: Algebraic number field and Coprime integers · See more »

Countable set

In mathematics, a countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers.

New!!: Algebraic number field and Countable set · See more »

Cyclotomic field

In number theory, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to, the field of rational numbers.

New!!: Algebraic number field and Cyclotomic field · See more »

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.

New!!: Algebraic number field and Dedekind domain · See more »

Dedekind zeta function

In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function (which is obtained in the case where K is the rational numbers Q).

New!!: Algebraic number field and Dedekind zeta function · See more »

Degree of a field extension

In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension.

New!!: Algebraic number field and Degree of a field extension · See more »

Determinant

In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.

New!!: Algebraic number field and Determinant · See more »

Dimension (vector space)

In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V over its base field.

New!!: Algebraic number field and Dimension (vector space) · See more »

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.

New!!: Algebraic number field and Dirichlet L-function · See more »

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 is a non-negative integer.

New!!: Algebraic number field and Dirichlet's theorem on arithmetic progressions · See more »

Dirichlet's unit theorem

In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet.

New!!: Algebraic number field and Dirichlet's unit theorem · See more »

Discrete valuation

In mathematics, a discrete valuation is an integer valuation on a field K; that is, a function satisfying the conditions for all x,y\in K. Note that often the trivial valuation which takes on only the values 0,\infty is explicitly excluded.

New!!: Algebraic number field and Discrete valuation · See more »

Discrete valuation ring

In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.

New!!: Algebraic number field and Discrete valuation ring · See more »

Discriminant

In algebra, the discriminant of a polynomial is a polynomial function of its coefficients, which allows deducing some properties of the roots without computing them.

New!!: Algebraic number field and Discriminant · See more »

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.

New!!: Algebraic number field and Discriminant of an algebraic number field · See more »

Division algebra

In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.

New!!: Algebraic number field and Division algebra · See more »

Duality (mathematics)

In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself.

New!!: Algebraic number field and Duality (mathematics) · See more »

Eisenstein integer

In mathematics, Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are complex numbers of the form where and are integers and is a primitive (hence non-real) cube root of unity.

New!!: Algebraic number field and Eisenstein integer · See more »

Euclidean domain

In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of the integers.

New!!: Algebraic number field and Euclidean domain · See more »

Euclidean vector

In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction.

New!!: Algebraic number field and Euclidean vector · See more »

Euler's totient function

In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to.

New!!: Algebraic number field and Euler's totient function · See more »

Fiber (mathematics)

In mathematics, the term fiber (or fibre in British English) can have two meanings, depending on the context.

New!!: Algebraic number field and Fiber (mathematics) · See more »

Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

New!!: Algebraic number field and Field (mathematics) · See more »

Field extension

In mathematics, and in particular, algebra, a field E is an extension field of a field F if E contains F and the operations of F are those of E restricted to F. Equivalently, F is a subfield of E. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.

New!!: Algebraic number field and Field extension · See more »

Field norm

In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield.

New!!: Algebraic number field and Field norm · See more »

Field of fractions

In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded.

New!!: Algebraic number field and Field of fractions · See more »

Field trace

In mathematics, the field trace is a particular function defined with respect to a finite field extension L/K, which is a ''K''-linear map from L onto K.

New!!: Algebraic number field and Field trace · See more »

Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.

New!!: Algebraic number field and Finite field · See more »

Finitely generated module

In mathematics, a finitely generated module is a module that has a finite generating set.

New!!: Algebraic number field and Finitely generated module · See more »

Frobenius endomorphism

In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic, an important class which includes finite fields.

New!!: Algebraic number field and Frobenius endomorphism · See more »

Function field of an algebraic variety

In algebraic geometry, the function field of an algebraic variety V consists of objects which are interpreted as rational functions on V. In classical algebraic geometry they are ratios of polynomials; in complex algebraic geometry these are meromorphic functions and their higher-dimensional analogues; in modern algebraic geometry they are elements of some quotient ring's field of fractions.

New!!: Algebraic number field and Function field of an algebraic variety · See more »

Functional equation

In mathematics, a functional equation is any equation in which the unknown represents a function.

New!!: Algebraic number field and Functional equation · See more »

Fundamental theorem of Galois theory

In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions.

New!!: Algebraic number field and Fundamental theorem of Galois theory · See more »

Galois cohomology

In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups.

New!!: Algebraic number field and Galois cohomology · See more »

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.

New!!: Algebraic number field and Galois group · See more »

Gaussian integer

In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers.

New!!: Algebraic number field and Gaussian integer · See more »

Gaussian rational

In mathematics, a Gaussian rational number is a complex number of the form p + qi, where p and q are both rational numbers.

New!!: Algebraic number field and Gaussian rational · See more »

Genus field

In algebraic number theory, the genus field G of an algebraic number field K is the maximal abelian extension of K which is obtained by composing an absolutely abelian field with K and which is unramified at all finite primes of K. The genus number of K is the degree and the genus group is the Galois group of G over K. If K is itself absolutely abelian, the genus field may be described as the maximal absolutely abelian extension of K unramified at all finite primes: this definition was used by Leopoldt and Hasse.

New!!: Algebraic number field and Genus field · See more »

Geometry

Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.

New!!: Algebraic number field and Geometry · See more »

Geometry of numbers

In number theory, the geometry of numbers studies convex bodies and integer vectors in n-dimensional space.

New!!: Algebraic number field and Geometry of numbers · See more »

Global field

In mathematics, a global field is a field that is either.

New!!: Algebraic number field and Global field · See more »

Glossary of algebraic geometry

This is a glossary of algebraic geometry.

New!!: Algebraic number field and Glossary of algebraic geometry · See more »

Group action

In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.

New!!: Algebraic number field and Group action · See more »

Group cohomology

In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology.

New!!: Algebraic number field and Group cohomology · See more »

Hasse principle

In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number.

New!!: Algebraic number field and Hasse principle · See more »

Hilbert class field

In algebraic number theory, the Hilbert class field E of a number field K is the maximal abelian unramified extension of K. Its degree over K equals the class number of K and the Galois group of E over K is canonically isomorphic to the ideal class group of K using Frobenius elements for prime ideals in K. In this context, the Hilbert class field of K is not just unramified at the finite places (the classical ideal theoretic interpretation) but also at the infinite places of K. That is, every real embedding of K extends to a real embedding of E (rather than to a complex embedding of E).

New!!: Algebraic number field and Hilbert class field · See more »

Homogeneous function

In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor.

New!!: Algebraic number field and Homogeneous function · See more »

Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.

New!!: Algebraic number field and Ideal (ring theory) · See more »

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.

New!!: Algebraic number field and Ideal class group · See more »

Image (mathematics)

In mathematics, an image is the subset of a function's codomain which is the output of the function from a subset of its domain.

New!!: Algebraic number field and Image (mathematics) · See more »

Imaginary unit

The imaginary unit or unit imaginary number is a solution to the quadratic equation.

New!!: Algebraic number field and Imaginary unit · See more »

Integer matrix

In mathematics, an integer matrix is a matrix whose entries are all integers.

New!!: Algebraic number field and Integer matrix · See more »

Integral domain

In mathematics, and specifically in abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.

New!!: Algebraic number field and Integral domain · See more »

Integrally closed

In mathematics, more specifically in abstract algebra, the concept of integrally closed has two meanings, one for groups and one for rings.

New!!: Algebraic number field and Integrally closed · See more »

Intermediate value theorem

In mathematical analysis, the intermediate value theorem states that if a continuous function, f, with an interval,, as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.

New!!: Algebraic number field and Intermediate value theorem · See more »

Invariant (mathematics)

In mathematics, an invariant is a property, held by a class of mathematical objects, which remains unchanged when transformations of a certain type are applied to the objects.

New!!: Algebraic number field and Invariant (mathematics) · See more »

Inverse element

In abstract algebra, the idea of an inverse element generalises concepts of a negation (sign reversal) in relation to addition, and a reciprocal in relation to multiplication.

New!!: Algebraic number field and Inverse element · See more »

Iwasawa theory

In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields.

New!!: Algebraic number field and Iwasawa theory · See more »

Kronecker–Weber theorem

In algebraic number theory, it can be shown that every cyclotomic field is an abelian extension of the rational number field Q. The Kronecker–Weber theorem provides a partial converse: every finite abelian extension of Q is contained within some cyclotomic field.

New!!: Algebraic number field and Kronecker–Weber theorem · See more »

Krull dimension

In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals.

New!!: Algebraic number field and Krull dimension · See more »

Kummer theory

In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field.

New!!: Algebraic number field and Kummer theory · See more »

Kurt Hensel

Kurt Wilhelm Sebastian Hensel (29 December 1861 – 1 June 1941) was a German mathematician born in Königsberg.

New!!: Algebraic number field and Kurt Hensel · See more »

L-function

In mathematics, an L-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects.

New!!: Algebraic number field and L-function · See more »

Limit of a sequence

As the positive integer n becomes larger and larger, the value n\cdot \sin\bigg(\frac1\bigg) becomes arbitrarily close to 1.

New!!: Algebraic number field and Limit of a sequence · See more »

Linear combination

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).

New!!: Algebraic number field and Linear combination · See more »

Linear function

In mathematics, the term linear function refers to two distinct but related notions.

New!!: Algebraic number field and Linear function · See more »

List of number fields with class number one

This is an incomplete list of number fields with class number 1.

New!!: Algebraic number field and List of number fields with class number one · See more »

Local analysis

In mathematics, the term local analysis has at least two meanings - both derived from the idea of looking at a problem relative to each prime number p first, and then later trying to integrate the information gained at each prime into a 'global' picture.

New!!: Algebraic number field and Local analysis · See more »

Local class field theory

In mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite residue field: hence every local field is isomorphic (as a topological field) to the real numbers R, the complex numbers C, a finite extension of the ''p''-adic numbers Qp (where p is any prime number), or a finite extension of the field of formal Laurent series Fq((T)) over a finite field Fq.

New!!: Algebraic number field and Local class field theory · See more »

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.

New!!: Algebraic number field and Local field · See more »

Local ring

In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime.

New!!: Algebraic number field and Local ring · See more »

Localization of a ring

In commutative algebra, localization is a systematic method of adding multiplicative inverses to a ring.

New!!: Algebraic number field and Localization of a ring · See more »

Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

New!!: Algebraic number field and Mathematics · See more »

Matrix (mathematics)

In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

New!!: Algebraic number field and Matrix (mathematics) · See more »

Matrix multiplication

In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring or even a semiring.

New!!: Algebraic number field and Matrix multiplication · See more »

Maximal ideal

In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals.

New!!: Algebraic number field and Maximal ideal · See more »

Minkowski's theorem

In mathematics, Minkowski's theorem is the statement that any convex set in \mathbb^n which is symmetric with respect to the origin and which has volume greater than 2^n contains a non-zero integer point.

New!!: Algebraic number field and Minkowski's theorem · See more »

Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli).

New!!: Algebraic number field and Modular arithmetic · See more »

Monic polynomial

In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1.

New!!: Algebraic number field and Monic polynomial · See more »

Monogenic field

In mathematics, a monogenic field is an algebraic number field K for which there exists an element a such that the ring of integers OK is the subring Z of K generated by a. Then OK is a quotient of the polynomial ring Z and the powers of a constitute a power integral basis.

New!!: Algebraic number field and Monogenic field · See more »

Multiplication

Multiplication (often denoted by the cross symbol "×", by a point "⋅", by juxtaposition, or, on computers, by an asterisk "∗") is one of the four elementary mathematical operations of arithmetic; with the others being addition, subtraction and division.

New!!: Algebraic number field and Multiplication · See more »

Noetherian ring

In mathematics, more specifically in the area of abstract algebra known as ring theory, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; that is, given any chain of left (or right) ideals: there exists an n such that: Noetherian rings are named after Emmy Noether.

New!!: Algebraic number field and Noetherian ring · See more »

Ordered pair

In mathematics, an ordered pair (a, b) is a pair of objects.

New!!: Algebraic number field and Ordered pair · See more »

Ostrowski's theorem

In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value.

New!!: Algebraic number field and Ostrowski's theorem · See more »

P-adic analysis

In mathematics, p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of ''p''-adic numbers.

New!!: Algebraic number field and P-adic analysis · See more »

P-adic number

In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems.

New!!: Algebraic number field and P-adic number · See more »

Polynomial

In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.

New!!: Algebraic number field and Polynomial · See more »

Prime element

In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials.

New!!: Algebraic number field and Prime element · See more »

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.

New!!: Algebraic number field and Prime ideal · See more »

Primitive element theorem

In field theory, the primitive element theorem or Artin's theorem on primitive elements is a result characterizing the finite degree field extensions that possess a primitive element, or simple extensions.

New!!: Algebraic number field and Primitive element theorem · See more »

Quadratic field

In algebraic number theory, a quadratic field is an algebraic number field K of degree two over Q, the rational numbers.

New!!: Algebraic number field and Quadratic field · See more »

Quadratic integer

In number theory, quadratic integers are a generalization of the integers to quadratic fields.

New!!: Algebraic number field and Quadratic integer · See more »

Ramification (mathematics)

In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two branches differing in sign.

New!!: Algebraic number field and Ramification (mathematics) · See more »

Ramification group

In number theory, more specifically in local class field theory, the ramification groups are a filtration of the Galois group of a local field extension, which gives detailed information on the ramification phenomena of the extension.

New!!: Algebraic number field and Ramification group · See more »

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.

New!!: Algebraic number field and Rational number · See more »

Ray class field

In mathematics, a ray class field is an abelian extension of a global field associated with a ray class group of ideal classes or idele classes.

New!!: Algebraic number field and Ray class field · See more »

Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

New!!: Algebraic number field and Real number · See more »

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.

New!!: Algebraic number field and Regular representation · See more »

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.

New!!: Algebraic number field and Residue (complex analysis) · See more »

Residue field

In mathematics, the residue field is a basic construction in commutative algebra.

New!!: Algebraic number field and Residue field · See more »

Resultant

In mathematics, the resultant of two polynomials is a polynomial expression of their coefficients, which is equal to zero if and only if the polynomials have a common root (possibly in a field extension), or, equivalently, a common factor (over their field of coefficients).

New!!: Algebraic number field and Resultant · See more »

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.

New!!: Algebraic number field and Richard Dedekind · See more »

Riemann surface

In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold.

New!!: Algebraic number field and Riemann surface · See more »

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.

New!!: Algebraic number field and Riemann zeta function · See more »

Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

New!!: Algebraic number field and Ring (mathematics) · See more »

Ring of integers

In mathematics, the ring of integers of an algebraic number field is the ring of all integral elements contained in.

New!!: Algebraic number field and Ring of integers · See more »

Root of unity

In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power.

New!!: Algebraic number field and Root of unity · See more »

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.

New!!: Algebraic number field and S-unit · See more »

Scalar multiplication

In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra).

New!!: Algebraic number field and Scalar multiplication · See more »

Scheme (mathematics)

In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x.

New!!: Algebraic number field and Scheme (mathematics) · See more »

Serge Lang

Serge Lang (May 19, 1927 – September 12, 2005) was a French-born American mathematician and activist.

New!!: Algebraic number field and Serge Lang · See more »

Set (mathematics)

In mathematics, a set is a collection of distinct objects, considered as an object in its own right.

New!!: Algebraic number field and Set (mathematics) · See more »

Sheaf (mathematics)

In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.

New!!: Algebraic number field and Sheaf (mathematics) · See more »

Simple extension

In field theory, a simple extension is a field extension which is generated by the adjunction of a single element.

New!!: Algebraic number field and Simple extension · See more »

Special values of L-functions

In mathematics, the study of special values of L-functions is a subfield of number theory devoted to generalising formulae such as the Leibniz formula for pi, namely by the recognition that expression on the left-hand side is also L(1) where L(s) is the Dirichlet L-function for the Gaussian field.

New!!: Algebraic number field and Special values of L-functions · See more »

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.

New!!: Algebraic number field and Spectrum of a ring · See more »

Splitting of prime ideals in Galois extensions

In mathematics, the interplay between the Galois group G of a Galois extension L of a number field K, and the way the prime ideals P of the ring of integers OK factorise as products of prime ideals of OL, provides one of the richest parts of algebraic number theory.

New!!: Algebraic number field and Splitting of prime ideals in Galois extensions · See more »

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.

New!!: Algebraic number field and Springer Science+Business Media · See more »

Square-free element

In mathematics, a square-free element is an element r of a unique factorization domain R that is not divisible by a non-trivial square.

New!!: Algebraic number field and Square-free element · See more »

Subring

In mathematics, a subring of R is a subset of a ring that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and which shares the same multiplicative identity as R. For those who define rings without requiring the existence of a multiplicative identity, a subring of R is just a subset of R that is a ring for the operations of R (this does imply it contains the additive identity of R).

New!!: Algebraic number field and Subring · See more »

Symmetric matrix

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose.

New!!: Algebraic number field and Symmetric matrix · See more »

Tate duality

In mathematics, Tate duality or Poitou–Tate duality is a duality theorem for Galois cohomology groups of modules over the Galois group of an algebraic number field or local field, introduced by and.

New!!: Algebraic number field and Tate duality · See more »

Tensor product of fields

In abstract algebra, the theory of fields lacks a direct product: the direct product of two fields, considered as a ring is never itself a field.

New!!: Algebraic number field and Tensor product of fields · See more »

Topological ring

In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps where R × R carries the product topology.

New!!: Algebraic number field and Topological ring · See more »

Topology

In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

New!!: Algebraic number field and Topology · See more »

Totally imaginary number field

In algebraic number theory, a number field is called totally imaginary (or totally complex) if it cannot be embedded in the real numbers.

New!!: Algebraic number field and Totally imaginary number field · See more »

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.

New!!: Algebraic number field and Totally real number field · See more »

Trace (linear algebra)

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e., where aii denotes the entry on the ith row and ith column of A. The trace of a matrix is the sum of the (complex) eigenvalues, and it is invariant with respect to a change of basis.

New!!: Algebraic number field and Trace (linear algebra) · See more »

Tuple

In mathematics, a tuple is a finite ordered list (sequence) of elements.

New!!: Algebraic number field and Tuple · See more »

Ultrametric space

In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d(x,z)\leq\max\left\.

New!!: Algebraic number field and Ultrametric space · See more »

Uncountable set

In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable.

New!!: Algebraic number field and Uncountable set · See more »

Unique factorization domain

In mathematics, a unique factorization domain (UFD) is an integral domain (a non-zero commutative ring in which the product of non-zero elements is non-zero) in which every non-zero non-unit element can be written as a product of prime elements (or irreducible elements), uniquely up to order and units, analogous to the fundamental theorem of arithmetic for the integers.

New!!: Algebraic number field and Unique factorization domain · See more »

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.

New!!: Algebraic number field and Unit (ring theory) · See more »

Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

New!!: Algebraic number field and Vector space · See more »

Weil's conjecture on Tamagawa numbers

In mathematics, the Weil conjecture on Tamagawa numbers is the statement that the Tamagawa number τ(G) of a simply connected simple algebraic group defined over a number field is 1.

New!!: Algebraic number field and Weil's conjecture on Tamagawa numbers · See more »

Zero divisor

In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero such that, or equivalently if the map from to that sends to is not injective.

New!!: Algebraic number field and Zero divisor · See more »

Zeros and poles

In mathematics, a zero of a function is a value such that.

New!!: Algebraic number field and Zeros and poles · See more »

Redirects here:

Algebraic number fields, Dedekind discriminant theorem, Degree of a number field, Degree of an algebraic number field, Number field, Number fields, Power basis.

References

[1] https://en.wikipedia.org/wiki/Algebraic_number_field

OutgoingIncoming
Hey! We are on Facebook now! »