# Euclidean domain

This article defines a property of integral domains, viz., a property that, given any integral domain, is either true or false for that.
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## Definition

### 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.

## Examples

### 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.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
uniquely Euclidean domain there is a Euclidean norm for which Euclidean division is unique. click here
Polynomial ring over a field it can be written as the polynomial ring $K[x]$ for a field $K$. click here

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
principal ideal domain integral domain that is a principal ideal ring click here
Bezout domain integral domain in which every finitely generated ideal is principal click here

### 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:

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

## Metaproperties

### Polynomial-closedness

This property of commutative unital rings is not closed under passing to the polynomial ring

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).