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- Euclidean_domain abstract "In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain (also called a Euclidean ring) is a commutative ring that can be endowed with a Euclidean function (explained below) which allows a suitable generalization of the Euclidean division of the integers. This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements. In particular, the greatest common divisor of any two elements exists and can be written as a linear combination of them (Bézout's identity). Also every ideal in a Euclidean domain is principal, which implies a suitable generalization of the fundamental theorem of arithmetic: every Euclidean domain is a unique factorization domain.It is important to compare the class of Euclidean domains with the larger class of principal ideal domains (PIDs). An arbitrary PID has much the same \"structural properties\" of a Euclidean domain (or, indeed, even of the ring of integers), but when an explicit algorithm for Euclidean division is known, one may use Euclidean algorithm and extended Euclidean algorithm to compute greatest common divisors and Bézout's identity. In particular, the existence of efficient algorithms for Euclidean division of integers and of polynomials in one variable over a field is of basic importance in computer algebra.So, given an integral domain R, it is often very useful to know that R has a Euclidean function: in particular, this implies that R is a PID. However, if there is no \"obvious\" Euclidean function, then determining whether R is a PID is generally a much easier problem than determining whether it is a Euclidean domain.Euclidean domains appear in the following chain of class inclusions: commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ finite fields".
- Euclidean_domain wikiPageID "10376".
- Euclidean_domain wikiPageLength "15969".
- Euclidean_domain wikiPageOutDegree "62".
- Euclidean_domain wikiPageRevisionID "704318243".
- Euclidean_domain wikiPageWikiLink Absolute_value.
- Euclidean_domain wikiPageWikiLink Abstract_algebra.
- Euclidean_domain wikiPageWikiLink Algebraic_number_field.
- Euclidean_domain wikiPageWikiLink Algorithm.
- Euclidean_domain wikiPageWikiLink Atomic_domain.
- Euclidean_domain wikiPageWikiLink Bxc3xa9zouts_identity.
- Euclidean_domain wikiPageWikiLink Category:All_articles_lacking_sources.
- Euclidean_domain wikiPageWikiLink Category:Commutative_algebra.
- Euclidean_domain wikiPageWikiLink Category:Euclid.
- Euclidean_domain wikiPageWikiLink Category:Ring_theory.
- Euclidean_domain wikiPageWikiLink Commutative_ring.
- Euclidean_domain wikiPageWikiLink Conjugate_element_(field_theory).
- Euclidean_domain wikiPageWikiLink Dedekind_domain.
- Euclidean_domain wikiPageWikiLink Dirichlets_unit_theorem.
- Euclidean_domain wikiPageWikiLink Discrete_valuation_ring.
- Euclidean_domain wikiPageWikiLink Division_algorithm.
- Euclidean_domain wikiPageWikiLink Eisenstein_integer.
- Euclidean_domain wikiPageWikiLink Euclidean_algorithm.
- Euclidean_domain wikiPageWikiLink Euclidean_division.
- Euclidean_domain wikiPageWikiLink Extended_Euclidean_algorithm.
- Euclidean_domain wikiPageWikiLink Field_(mathematics).
- Euclidean_domain wikiPageWikiLink Field_extension.
- Euclidean_domain wikiPageWikiLink Field_norm.
- Euclidean_domain wikiPageWikiLink Formal_power_series.
- Euclidean_domain wikiPageWikiLink Fundamental_theorem_of_arithmetic.
- Euclidean_domain wikiPageWikiLink Gaussian_integer.
- Euclidean_domain wikiPageWikiLink Generalized_Riemann_hypothesis.
- Euclidean_domain wikiPageWikiLink Greatest_common_divisor.
- Euclidean_domain wikiPageWikiLink Ideal_(ring_theory).
- Euclidean_domain wikiPageWikiLink Ideal_class_group.
- Euclidean_domain wikiPageWikiLink Integer.
- Euclidean_domain wikiPageWikiLink Integral_domain.
- Euclidean_domain wikiPageWikiLink Mathematics.
- Euclidean_domain wikiPageWikiLink Natural_number.
- Euclidean_domain wikiPageWikiLink Noetherian_ring.
- Euclidean_domain wikiPageWikiLink Polynomial_ring.
- Euclidean_domain wikiPageWikiLink Principal_ideal.
- Euclidean_domain wikiPageWikiLink Principal_ideal_domain.
- Euclidean_domain wikiPageWikiLink Quadratic_field.
- Euclidean_domain wikiPageWikiLink Ring_of_integers.
- Euclidean_domain wikiPageWikiLink Ring_theory.
- Euclidean_domain wikiPageWikiLink Root_of_unity.
- Euclidean_domain wikiPageWikiLink Subclass_(set_theory).
- Euclidean_domain wikiPageWikiLink Symbolic_computation.
- Euclidean_domain wikiPageWikiLink Unique_factorization_domain.
- Euclidean_domain wikiPageWikiLink Valuation_(algebra).
- Euclidean_domain wikiPageWikiLinkText "EDs".
- Euclidean_domain wikiPageWikiLinkText "Euclidean domain".
- Euclidean_domain wikiPageWikiLinkText "Euclidean domain#Definition".
- Euclidean_domain wikiPageWikiLinkText "Euclidean domain#Norm-Euclidean fields".
- Euclidean_domain wikiPageWikiLinkText "Euclidean function".
- Euclidean_domain wikiPageWikiLinkText "Euclidean".
- Euclidean_domain wikiPageWikiLinkText "Euclidean_domain".
- Euclidean_domain wikiPageUsesTemplate Template:=.
- Euclidean_domain wikiPageUsesTemplate Template:Brackets.
- Euclidean_domain wikiPageUsesTemplate Template:Citation_needed.
- Euclidean_domain wikiPageUsesTemplate Template:Commutative_ring_classes.
- Euclidean_domain wikiPageUsesTemplate Template:Math.
- Euclidean_domain wikiPageUsesTemplate Template:OEIS.
- Euclidean_domain subject Category:All_articles_lacking_sources.
- Euclidean_domain subject Category:Commutative_algebra.
- Euclidean_domain subject Category:Euclid.
- Euclidean_domain subject Category:Ring_theory.
- Euclidean_domain hypernym Ring.
- Euclidean_domain type AnatomicalStructure.
- Euclidean_domain type Scientist.
- Euclidean_domain type Scientist.
- Euclidean_domain comment "In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain (also called a Euclidean ring) is a commutative ring that can be endowed with a Euclidean function (explained below) which allows a suitable generalization of the Euclidean division of the integers.".
- Euclidean_domain label "Euclidean domain".
- Euclidean_domain sameAs Q867345.
- Euclidean_domain sameAs Anell_euclidià.
- Euclidean_domain sameAs Eukleidovský_obor.
- Euclidean_domain sameAs Euklidischer_Ring.
- Euclidean_domain sameAs Ευκλείδεια_περιοχή.
- Euclidean_domain sameAs Dominio_euclídeo.
- Euclidean_domain sameAs Anneau_euclidien.
- Euclidean_domain sameAs חוג_אוקלידי.
- Euclidean_domain sameAs Euklideszi_gyűrű.
- Euclidean_domain sameAs Dominio_euclideo.
- Euclidean_domain sameAs ユークリッド環.
- Euclidean_domain sameAs 유클리드_정역.
- Euclidean_domain sameAs Euclidisch_domein.
- Euclidean_domain sameAs Euklidsk_ring.
- Euclidean_domain sameAs Dziedzina_Euklidesa.
- Euclidean_domain sameAs Domínio_euclidiano.
- Euclidean_domain sameAs m.02tb6.
- Euclidean_domain sameAs Евклидово_кольцо.
- Euclidean_domain sameAs Euklidiskt_område.
- Euclidean_domain sameAs Евклідове_кільце.
- Euclidean_domain sameAs Q867345.
- Euclidean_domain sameAs 歐幾里得整環.
- Euclidean_domain wasDerivedFrom Euclidean_domain?oldid=704318243.
- Euclidean_domain isPrimaryTopicOf Euclidean_domain.