Euclidean domain

Symbol-free definition
An integral domain is said to be Euclidean if it admits a Euclidean norm.

Definition with symbols
An integral domain $$R$$ is termed a Euclidean domain if there exists a function $$N$$ from the set of nonzero elements of $$R$$ to the set of nonnegative integers satisfying the following properties:


 * $$N(x) = 0$$ if and only if $$x$$ is a unit
 * Given nonzero $$a$$ and $$b$$ in $$R$$, there exist $$q$$ and $$r$$ such that $$a = qb + r$$ and either $$r = 0$$ or $$N(r) < N(b)$$.

We call $$a$$ the dividend, $$b$$ the divisor, $$q$$ the quotient and $$r$$ the remainder.

Such a function $$N$$ is called a Euclidean norm on $$R$$.

Caveats

 * The definition of Euclidean norm does not require the ring to be an integral domain. A commutative unital ring that admits a Euclidean norm is termed a Euclidean ring.
 * The definition of Euclidean domain does not require that $$q$$ and $$r$$ be uniquely determined from $$a$$ and $$b$$. If $$q$$ and $$r$$ are uniquely determined from $$a$$ and $$b$$, the integral domain is termed a uniquely Euclidean domain.

Standard examples

 * The ring of rational integers $$\mathbb{Z}$$ is a Euclidean domain with Euclidean norm defined by the absolute value.
 * The polynomial ring over a field $$k[x]$$ is a Euclidean domain with Euclidean norm defined by the degree of a polynomial. This is, in fact a uniquely Euclidean norm. and hence the polynomial ring over a field is a uniquely Euclidean domain.

Other examples

 * The ring of Gaussian integers $$\mathbb{Z}[i]$$ is a Euclidean domain with Euclidean norm equal to the norm in the sense of a quadratic integer ring.
 * A quadratic integer ring, or more generally, a ring of integers in a number field, is termed norm-Euclidean ring of integers in a number field if it is Euclidean with respect to the algebraic norm. Since there is a correspondence between number fields and their rings of integers, we often abuse language and say that the number field itself is norm-Euclidean.
 * Any discrete valuation ring is a Euclidean domain where the norm of an element is given by the largest power of the irreducible that divides it. For instance, the formal power series ring over a field is a Euclidean domain, where the norm of a formal power series is the smallest $$n$$ for which the coefficient of $$x^n$$ that is nonzero.

Pathological examples
On a field, any norm function is Euclidean. This is because we can always choose a quotient so that the remainder is zero.

Properties of Euclidean norms
Euclidean norms can in general be very weirdly behaved, but some Euclidean norms are good. For a complete list of properties of Euclidean norms (i.e., properties against which a given Euclidean norm can be tested), refer:

Category:Properties of Euclidean norms

Here are some important properties that most typical Euclidean norms satisfy:


 * Multiplicatively monotone Euclidean norm

Metaproperties
The polynomial ring over a Euclidean domain need not be a Euclidean domain. One example is the polynomial ring with integer coefficients, which is not a Euclidean domain; another example is the polynomial ring in two variables over a field (which can be viewed as the polynomial ring in one variable, over the polynomial ring over a field).