Faster access than browser!
35 relations: Algebraic number field, Algebraic number theory, Basis (linear algebra), Cambridge University Press, Cyclotomic field, Dedekind domain, Degree of a field extension, Dirichlet's unit theorem, Field (mathematics), Finitely generated abelian group, Free module, Fundamental theorem of arithmetic, Fundamental unit (number theory), Integer, Integral element, Irreducible element, Local field, Mathematics, Modular arithmetic, Module (mathematics), Monic polynomial, Order (ring theory), P-adic number, Prime ideal, Prime number, Quadratic field, Quadratic integer, Rational number, Ring (mathematics), Root of unity, Square-free integer, Subring, Torsion subgroup, Unit (ring theory), Zero of a function.
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!!: Ring of integers 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!!: Ring of integers and Algebraic number theory · See more »
Basis (linear algebra)
In mathematics, a set of elements (vectors) in a vector space V is called a basis, or a set of, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.
New!!: Ring of integers and Basis (linear algebra) · See more »
Cambridge University Press
Cambridge University Press (CUP) is the publishing business of the University of Cambridge.
New!!: Ring of integers and Cambridge University Press · 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!!: Ring of integers 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!!: Ring of integers and Dedekind domain · 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!!: Ring of integers and Degree of a field extension · 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!!: Ring of integers and Dirichlet's unit theorem · 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!!: Ring of integers and Field (mathematics) · See more »
Finitely generated abelian group
In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements x1,..., xs in G such that every x in G can be written in the form with integers n1,..., ns.
New!!: Ring of integers and Finitely generated abelian group · See more »
Free module
In mathematics, a free module is a module that has a basis – that is, a generating set consisting of linearly independent elements.
New!!: Ring of integers and Free module · See more »
Fundamental theorem of arithmetic
In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors.
New!!: Ring of integers and Fundamental theorem of arithmetic · See more »
Fundamental unit (number theory)
In algebraic number theory, a fundamental unit is a generator (modulo the roots of unity) for the unit group of the ring of integers of a number field, when that group has rank 1 (i.e. when the unit group modulo its torsion subgroup is infinite cyclic).
New!!: Ring of integers and Fundamental unit (number theory) · See more »
Integer
An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").
New!!: Ring of integers and Integer · See more »
Integral element
In commutative algebra, an element b of a commutative ring B is said to be integral over A, a subring of B, if there are n ≥ 1 and a_j \in A such that That is to say, b is a root of a monic polynomial over A. If every element of B is integral over A, then it is said that B is integral over A, or equivalently B is an integral extension of A. If A, B are fields, then the notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial).
New!!: Ring of integers and Integral element · See more »
Irreducible element
In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units.
New!!: Ring of integers and Irreducible element · 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!!: Ring of integers and Local field · See more »
Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
New!!: Ring of integers and Mathematics · 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!!: Ring of integers and Modular arithmetic · See more »
Module (mathematics)
In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra.
New!!: Ring of integers and Module (mathematics) · 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!!: Ring of integers and Monic polynomial · See more »
Order (ring theory)
In mathematics, an order in the sense of ring theory is a subring \mathcal of a ring A, such that.
New!!: Ring of integers and Order (ring theory) · 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!!: Ring of integers and P-adic number · 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!!: Ring of integers and Prime ideal · See more »
Prime number
A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.
New!!: Ring of integers and Prime number · 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!!: Ring of integers and Quadratic field · See more »
Quadratic integer
In number theory, quadratic integers are a generalization of the integers to quadratic fields.
New!!: Ring of integers and Quadratic integer · 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!!: Ring of integers and Rational number · See more »
Ring (mathematics)
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.
New!!: Ring of integers and Ring (mathematics) · 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!!: Ring of integers and Root of unity · See more »
Square-free integer
In mathematics, a square-free integer is an integer which is divisible by no perfect square other than 1.
New!!: Ring of integers and Square-free integer · 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!!: Ring of integers and Subring · See more »
Torsion subgroup
In the theory of abelian groups, the torsion subgroup AT of an abelian group A is the subgroup of A consisting of all elements that have finite order (the torsion elements of A).
New!!: Ring of integers and Torsion subgroup · 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!!: Ring of integers and Unit (ring theory) · See more »
Zero of a function
In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation f(x).
New!!: Ring of integers and Zero of a function · See more »
Redirects here:
Algebraic number ring, Integer ring, Integral basis, Number ring, Ring of integer, Rings of integers.
References
[1] https://en.wikipedia.org/wiki/Ring_of_integers
