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...

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...

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...

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 ⊃...

space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces...

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...

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

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...

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...

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...

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

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...

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...

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

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...

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...

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

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

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...

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...

GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields A division ring is a ring...

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

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...