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 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 ![]() ![]() |
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).