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...

In mathematics, the Euclidean algorithm, or Euclid's algorithm, is an efficient method for computing the greatest common divisor (GCD) of two integers...

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...

In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a...

integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed...

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...

integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed...

⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃...

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...

many properties with integers: they form a Euclidean domain, and have thus a Euclidean division and a Euclidean algorithm; this implies unique factorization...

arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest...

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 division similar to that of integers. Every Euclidean domain is a principal ideal domain, and thus a UFD. In a Euclidean domain, Euclidean division...

integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields...

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...

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...

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...

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...

⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃...

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...

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...

{\displaystyle \mathbb {Z} } is a Euclidean domain. This implies that Z {\displaystyle \mathbb {Z} } is a principal ideal domain, and any positive integer can...

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...

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...

their factor rings. Summary: Euclidean domain ⊂ principal ideal domain ⊂ unique factorization domain ⊂ integral domain ⊂ commutative ring. Algebraic...

mathematical structure. Those in which a Euclidean division (with remainder) is defined are called Euclidean domains and include polynomial rings in one indeterminate...

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...

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...

division, a more concise method of performing Euclidean polynomial division Ruffini's rule Euclidean domain Gröbner basis Greatest common divisor of two...

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...