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Commutative ring with a Euclidean division
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
Euclidean_domain
Division with remainder of integers
to integers, Euclidean division and the division theorem can be generalized to univariate polynomials over a field and to Euclidean domains. In the case
Euclidean_division
Algorithm for computing greatest common divisors
variable. This led to modern abstract algebraic notions such as Euclidean domains. The Euclidean algorithm calculates the greatest common divisor (GCD) of two
Euclidean_algorithm
Algebraic structure
Principal ideal domains are Noetherian, they are integrally closed, they are unique factorization domains and Dedekind domains. All Euclidean domains and all
Principal_ideal_domain
Commutative ring with no zero divisors other than zero
⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃
Integral_domain
Complex number whose mapping on a coordinate plane produces a triangular lattice
Eisenstein integers of norm 1. The ring of Eisenstein integers forms a Euclidean domain whose norm N is given by the square modulus, as above: N ( a + b ω
Eisenstein_integer
Method for computing the relation of two integers with their greatest common divisor
arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest
Extended_Euclidean_algorithm
Algebraic structure
integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed
Integrally_closed_domain
Complex number whose real and imaginary parts are both integers
many properties with integers: they form a Euclidean domain, and thus have a Euclidean division and a Euclidean algorithm; this implies unique factorization
Gaussian_integer
Algebraic construction
is a Euclidean domain. The ring of integers of an algebraic number field is the unique maximal order in the field. It is always a Dedekind domain. The
Ring_of_integers
About simultaneous modular congruences
Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely
Chinese_remainder_theorem
function on an integral domain that generalises the notion of a Euclidean function on Euclidean domains. Let R be an integral domain and g : R → Z≥0 be a
Dedekind–Hasse_norm
Topics referred to by the same term
two numbers Euclidean domain, a ring in which Euclidean division may be defined, which allows Euclid's lemma to be true and the Euclidean algorithm and
Euclidean
Type of integral domain
integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields
Unique_factorization_domain
Topics referred to by the same term
elements Bézout domain, an integral domain in which the sum of two principal ideals is again a principal ideal Euclidean domain, an integral domain which allows
Domain
Root of a quadratic polynomial with a unit leading coefficient
real quadratic integers that is a principal ideal domain is also a Euclidean domain for some Euclidean function, which can indeed differ from the usual
Quadratic_integer
Fundamental space of geometry
space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces
Euclidean_space
Algebraic structure with addition and multiplication
⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃
Ring_(mathematics)
Greatest common divisor of polynomials
rings for which such a theorem exists are called Euclidean domains. Like for the integers, the Euclidean division of the polynomials may be computed by
Polynomial greatest common divisor
Polynomial_greatest_common_divisor
Number in {..., –2, –1, 0, 1, 2, ...}
{\displaystyle \mathbb {Z} } is a Euclidean domain. This implies that Z {\displaystyle \mathbb {Z} } is a principal ideal domain, and any positive integer
Integer
Arithmetic operation
mathematical structure. Those in which a Euclidean division (with remainder) is defined are called Euclidean domains and include polynomial rings in one indeterminate
Division_(mathematics)
Algebraic structure
either r = 0 or deg(r) < deg(b). This makes K[X] a Euclidean domain. However, most other Euclidean domains (except integers) do not have any property of uniqueness
Polynomial_ring
(Mathematical) decomposition into a product
an integral domain on which is defined a Euclidean division similar to that of integers. Every Euclidean domain is a principal ideal domain, and thus a
Factorization
Mathematical concept
polynomial ring R[x] is a principal ideal domain and, more importantly to our discussion here, a Euclidean domain. It can be shown that the degree of a polynomial
Degree_of_a_polynomial
Branch of algebra
their factor rings. Summary: Euclidean domain ⊂ principal ideal domain ⊂ unique factorization domain ⊂ integral domain ⊂ commutative ring. Algebraic
Ring_theory
Geometric model of the physical space
domain), a solid figure. Technically, a tuple of n numbers can be understood as the Cartesian coordinates of a location in a n-dimensional Euclidean space
Three-dimensional_space
Algebraic structure
⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃
Commutative_ring
Type of binary relation
Similarly, the domain of a left Euclidean relation is a subset of its range, and the restriction of a left Euclidean relation to its domain is an equivalence
Euclidean_relation
Mathematical structure with greatest common divisors
⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃
GCD_domain
Group of matrices with determinant 1
group over a field or a Euclidean domain is generated by transvections, and the stable special linear group over a Dedekind domain is generated by transvections
Special_linear_group
Largest integer that divides given integers
integral domains. However, if R is a unique factorization domain or any other GCD domain, then any two elements have a GCD. If R is a Euclidean domain in which
Greatest_common_divisor
Integers have unique prime factorizations
structures that are called unique factorization domains and include principal ideal domains, Euclidean domains, and polynomial rings over a field. However
Fundamental theorem of arithmetic
Fundamental_theorem_of_arithmetic
Algorithm for division of polynomials
x^{3}-12x^{2}+24=(x-3)(x^{2}-9x-27)-57} . Polynomial remainder theorem Ruffini's rule Euclidean domain Gröbner basis Greatest common divisor of two polynomials S. Barnard
Polynomial_long_division
Domain in a Euclidean space whose boundary is sufficiently regular
In mathematics, a Lipschitz domain (or domain with Lipschitz boundary) is a domain in Euclidean space whose boundary is "sufficiently regular" in the
Lipschitz_domain
Equations of motion for viscous fluids
fundamental equation of hydraulics. The domain for these equations is commonly a 3 or fewer dimensional Euclidean space, for which an orthogonal coordinate
Navier–Stokes_equations
Generalization of vector spaces from fields to rings
realm of modules over a "well-behaved" ring, such as a principal ideal domain. However, modules can be quite a bit more complicated than vector spaces;
Module_(mathematics)
Mathematical operation
derivative detects multiple roots. If R is a field then R[x] is a Euclidean domain, and in this situation we can define multiplicity of roots; for every
Formal_derivative
Condition under which an odd prime is a sum of two squares
that the Gaussian integers are a unique factorization domain (because they are a Euclidean domain). Since p ∈ Z does not divide either of the Gaussian
Fermat's theorem on sums of two squares
Fermat's_theorem_on_sums_of_two_squares
Algebraic structure where all polynomials have roots
⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃
Algebraically_closed_field
Algebraic structure used in analysis
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Lie_algebra
Algebraic ring without a multiplicative identity
⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃
Rng_(algebra)
Submodule of a mathematical ring
positive element, as a consequence of Euclidean division, so Z {\displaystyle \mathbb {Z} } is a principal ideal domain. The set of all polynomials with real
Ideal_(ring_theory)
Amount left over after computation
division is valid. The rings for which such a theorem exists are called Euclidean domains, but in this generality, uniqueness of the quotient and remainder
Remainder
Free object in the category of associative algebras
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Free_algebra
Algebraic structure
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Semifield
Algorithm for Euclidean division of polynomials
In algebra, synthetic division is a method for manually performing Euclidean division of polynomials, with less writing and fewer calculations than long
Synthetic_division
Abstract algebra concept
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions
Field_of_fractions
Topics referred to by the same term
a field Norm function, a term in the study of Euclidean domains, sometimes used in place of "Euclidean function" Norm (descriptive set theory), a map
Norm
Mathematical structure in abstract algebra
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
*-algebra
Analogue of a prime number in a commutative ring
R {\displaystyle p\in R} often restrict R to be an integral domain or a Euclidean domain, or may add the additional requirement that p is not a zero-divisor
Prime_element
Finite extension of the rationals
{\displaystyle {\mathcal {O}}_{\mathbf {Q} ({\sqrt {-5}})}} . Euclidean domains are unique factorization domains: For example Z [ i ] {\displaystyle \mathbf {Z} [i]}
Algebraic_number_field
Result of commutative algebra
is some ideal. Since k [ y ] {\displaystyle k[y]} is a PID (it is a Euclidean domain), I = ( f ) {\displaystyle I=(f)} . If f = 0 {\displaystyle f=0} we
Noether_normalization_lemma
Type of topological space
manifold is a topological space that locally resembles real n-dimensional Euclidean space. Topological manifolds are an important class of topological spaces
Topological_manifold
Type of mathematical expression
This is called Euclidean division, division with remainder or polynomial long division and shows that the ring F[x] is a Euclidean domain. Analogously,
Polynomial
Set of all things that may be the input of a mathematical function
In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by dom ( f ) {\displaystyle \operatorname
Domain_of_a_function
Branch of algebra that studies commutative rings
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Commutative_algebra
Fraction with denominator a power of two
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Dyadic_rational
Property of point sets in Euclidean spaces
geometry, a set S {\displaystyle S} in the Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is called a star domain (or star-convex set, star-shaped set
Star_domain
Description in spectral theory
closed manifolds. Robert Seeley extended this to include certain Euclidean domains in 1978. In 1975, Hans Duistermaat and Victor Guillemin proved the
Weyl_law
Structure-preserving function between two rings
is a maximal ideal of R. If R and S are commutative and S is an integral domain, then ker(f) is a prime ideal of R. If R and S are commutative, S is a field
Ring_homomorphism
Subset of a ring that forms a ring itself
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Subring
Branch of functional analysis
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Operator_algebra
Set without nontrivial polynomial equalities
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Algebraic_independence
Algebraic ring that need not have additive negative elements
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Semiring
theorem Euclidean domain Euclidean field Euclidean group Euclidean geometry Non-Euclidean geometry Euclid's formula Euclidean distance Euclidean distance
List of things named after Euclid
List_of_things_named_after_Euclid
Integer side lengths of a right triangle
(m+ni)^{2}=(m^{2}-n^{2})+2mni.} Using the facts that the Gaussian integers are a Euclidean domain and that for a Gaussian integer p | p | 2 {\displaystyle |p|^{2}} is
Pythagorean_triple
Branch of number theory
Diophantine equations, such as 26x + 65y = 13, may be found using the Euclidean algorithm (c. 5th century BC). Diophantus's major work was the Arithmetica
Algebraic_number_theory
Submodule of fractions in abstract algebra
of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals
Fractional_ideal
Concept in abstract algebra
{\displaystyle \nu } also makes any discrete valuation ring into a Euclidean domain.[citation needed] Every discrete valuation ring, being a local ring
Discrete_valuation_ring
Every natural number can be represented as the sum of four integer squares
Hurwitz quaternions is not commutative, hence it is not an actual Euclidean domain, and it does not have unique factorization in the usual sense. Nevertheless
Lagrange's four-square theorem
Lagrange's_four-square_theorem
Unique ring consisting of one element
two advantages to considering it not to be a domain. First, this agrees with the definition that a domain is a ring in which 0 is the only zero divisor
Zero_ring
Algebra over a field where binary multiplication is not necessarily associative
include Lie algebras, Jordan algebras, the octonions, and three-dimensional Euclidean space equipped with the cross product operation. Since it is not assumed
Non-associative_algebra
Operations on ordinals that extend classical arithmetic
a ring. Hence the ordinals are not a Euclidean domain, since they are not even a ring; furthermore the Euclidean "norm" would be ordinal-valued using
Ordinal_arithmetic
Algebraic structure
converse does not hold: every right Ore domain is a right Goldie domain, and hence so is every commutative integral domain. A consequence of Goldie's theorem
Noncommutative_ring
Algebraic operation on coordinate vectors
numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the scalar product of two vectors is the dot product of their
Dot_product
follows easily from the fact that Z {\displaystyle \mathbb {Z} } is a Euclidean domain. The Whitehead group of a free abelian group is trivial, a 1964 result
Whitehead_torsion
Reduction of a ring by one of its ideals
distinct from the "quotient field", or field of fractions, of an integral domain as well as from the more general "rings of quotients" obtained by localization
Quotient_ring
Relating two numbers and their greatest common divisor
ideal domains. If a and b are not both zero and one pair of Bézout coefficients (x, y) has been computed (for example, using the extended Euclidean algorithm)
Bézout's_identity
Mathematical term in group theory
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Prüfer_group
Field of electrical engineering
processing generalizes signal processing tasks to signals living on non-Euclidean domains whose structure can be captured by a weighted graph. Graph signal
Signal_processing
Ring built from other rings (mathematics)
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Product_of_rings
Elements taken to zero by a homomorphism
domain of the homomorphism become related in the image. A homomorphism is a function that preserves the underlying algebraic structure in the domain to
Kernel_(algebra)
Study of numbers that are not solutions of polynomials with rational coefficients
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Transcendental_number_theory
Ring that is also a vector space or a module
analog of Levi's theorem for Lie algebras. Let R be a Noetherian integral domain with field of fractions K (for example, they can be Z, Q). A lattice L in
Associative_algebra
Topological space that locally resembles Euclidean space
mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n {\displaystyle n} -dimensional
Manifold
Tensor product of algebras over a field; itself another algebra
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Tensor_product_of_algebras
Special case of colimit in category theory
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Direct_limit
Subject area in mathematics
K_{1}(A)\cong A^{\times }\oplus SK_{1}(A)} . When A {\displaystyle A} is a Euclidean domain (e.g. a field, or the integers) S K 1 ( A ) {\displaystyle SK_{1}(A)}
Algebraic_K-theory
Algebra based on a vector space with a quadratic form
space V be real three-dimensional space R3, and the quadratic form be Euclidean. Then, for v, w in R3 we have the bilinear form (or scalar product) v
Clifford_algebra
Method in computational algebra
the ring of polynomials over a field is a Euclidean domain, we may compute these GCDs using the Euclidean algorithm. With some abstract algebra, the
Berlekamp's_algorithm
Rational numbers with root 5 added
[\varphi ]} is a Euclidean domain with the absolute value of the norm as its Euclidean function, meaning a version of the Euclidean algorithm can be used
Golden_field
Infinite sum that is considered independently from any notion of convergence
version of the Hilbert basis theorem), if R {\displaystyle R} is an integral domain, then so is R [ [ X ] ] {\displaystyle R[[X]]} , and if K {\displaystyle
Formal_power_series
Australian and American mathematician (born 1975)
conjecture in the 1970s, positing a tile-based characterisation of those Euclidean domains for which a Fourier ensemble provides a basis of L2. Tao resolved
Terence_Tao
Type of non-Euclidean geometry
geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: For any given line R
Hyperbolic_geometry
Ring ideal generated by a single element of the ring
of generators it follows that I {\displaystyle I} is principal. Any Euclidean domain is a PID; the algorithm used to calculate greatest common divisors
Principal_ideal
Conditions under which the congruence x^3 equals p (mod q) is solvable
[\omega ]=\left\{a+b\omega \ :\ a,b\in \mathbb {Z} \right\}.} This is a Euclidean domain with the norm function given by: N ( a + b ω ) = a 2 − a b + b 2 .
Cubic_reciprocity
Partition of space by hyperplanes
specialized to be all value q, then this is called the q-matrix (over the Euclidean domain Q [ q ] {\displaystyle \mathbb {Q} [q]} ) for the arrangement and much
Arrangement_of_hyperplanes
rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite
Free product of associative algebras
Free_product_of_associative_algebras
Commutative algebra studies commutative rings, their ideals, and modules over such rings
Polynomial ring Integral domain Boolean algebra (structure) Principal ideal domain Euclidean domain Unique factorization domain Dedekind domain Nilpotent elements
List of commutative algebra topics
List_of_commutative_algebra_topics
Concept in abstract algebra
contains non-zero polynomials, but as K [ X ] {\displaystyle K[X]} is a euclidean domain, it contains a unique polynomial p {\displaystyle p} with minimal degree
Algebraic_element
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EUCLIDEAN DOMAIN
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EUCLIDEAN DOMAIN
EUCLIDEAN DOMAIN
EUCLIDEAN DOMAIN
EUCLIDEAN DOMAIN
EUCLIDEAN DOMAIN
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