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...
19 KB (2,440 words) - 15:00, 29 May 2024
In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers...
123 KB (15,118 words) - 12:05, 9 July 2024
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...
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In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a...
16 KB (2,227 words) - 23:32, 17 March 2024
integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed...
10 KB (1,446 words) - 11:53, 24 June 2024
Eisenstein integer (section Euclidean domain)
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...
12 KB (1,643 words) - 13:40, 25 July 2024
⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃...
20 KB (3,124 words) - 14:07, 23 June 2024
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...
42 KB (7,184 words) - 16:57, 12 April 2024
integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed...
12 KB (1,924 words) - 02:04, 15 July 2024
Gaussian integer (section Euclidean division)
many properties with integers: they form a Euclidean domain, and have thus a Euclidean division and a Euclidean algorithm; this implies unique factorization...
35 KB (4,795 words) - 03:23, 20 December 2023
space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces...
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arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest...
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Factorization (section Unique factorization domains)
Euclidean division similar to that of integers. Every Euclidean domain is a principal ideal domain, and thus a UFD. In a Euclidean domain, Euclidean division...
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integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields...
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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...
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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...
2 KB (317 words) - 15:35, 3 March 2023
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...
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Greatest common divisor (section Euclidean algorithm)
and more generally this is true in GCD domains. If R is a Euclidean domain in which euclidean division is given algorithmically (as is the case for instance...
35 KB (4,674 words) - 11:27, 8 May 2024
⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃...
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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...
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{\displaystyle \mathbb {Z} } is a Euclidean domain. This implies that Z {\displaystyle \mathbb {Z} } is a principal ideal domain, and any positive integer can...
34 KB (3,941 words) - 09:33, 28 June 2024
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...
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Polynomial greatest common divisor (redirect from Euclidean division 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...
52 KB (7,865 words) - 14:33, 2 February 2024
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...
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their factor rings. Summary: Euclidean domain ⊂ principal ideal domain ⊂ unique factorization domain ⊂ integral domain ⊂ commutative ring. Algebraic...
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Division (mathematics) (section Euclidean division)
mathematical structure. Those in which a Euclidean division (with remainder) is defined are called Euclidean domains and include polynomial rings in one indeterminate...
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Three-dimensional space (redirect from Euclidean 3-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...
34 KB (4,829 words) - 17:02, 29 May 2024
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...
51 KB (8,173 words) - 20:35, 14 June 2024
Ring (mathematics) (section Domains)
GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields A division ring is a ring...
99 KB (13,682 words) - 20:28, 25 July 2024
This is called Euclidean division, division with remainder or polynomial long division and shows that the ring F[x] is a Euclidean domain. Analogously,...
60 KB (8,176 words) - 23:48, 29 June 2024