# 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.

View other properties of integral domains | View all properties of commutative unital ringsVIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions

## Definition

### Symbol-free definition

An integral domain is said to be **Euclidean** if it admits a Euclidean norm.

### Definition with symbols

An integral domain is termed a **Euclidean domain** if there exists a function from the set of nonzero elements of to the set of nonnegative integers satisfying the following properties:

- if and only if is a unit
- Given nonzero and in , there exist and such that and either or .

We call the *dividend*, the *divisor*, the *quotient* and the *remainder*.

Such a function is called a Euclidean norm on .

### 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 and be uniquely determined from and . If and are uniquely determined from and , the integral domain is termed a uniquely Euclidean domain.

## Examples

### Standard examples

- The ring of rational integers is a Euclidean domain with Euclidean norm defined by the absolute value.
*For full proof, refer: Ring of integers is Euclidean with norm equal to absolute value* - The polynomial ring over a field 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.*For full proof, refer: Polynomial ring over a field is uniquely Euclidean with norm equal to degree*

### Other examples

- The ring of Gaussian integers is a Euclidean domain with Euclidean norm equal to the norm in the sense of a quadratic integer ring.
*For full proof, refer: Ring of Gaussian integers is norm-Euclidean* - 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 for which the coefficient of 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.

## 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 for a field . | click here |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

multi-stage Euclidean domain | click here | |||

principal ideal domain | integral domain that is a principal ideal ring | Euclidean implies PID | PID not implies Euclidean | click here |

Bezout domain | integral domain in which every finitely generated ideal is principal | click here | ||

unique factorization domain | click here | |||

Dedekind domain | Noetherian, normal, one-dimensional domain | click here | ||

Noetherian domain | integral domain and every ideal is finitely generated | click here | ||

Noetherian ring | every ideal is finitely generated | 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:

Category:Properties of Euclidean norms

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).