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

Standard examples

• The ring of integers $\mathbb{Z}$ 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 $k[x]$ 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. For full proof, refer: Polynomial ring over a field is uniquely Euclidean with norm equal to degree

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. 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 $n$ 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.

Relation with other properties

Stronger properties

• Uniquely Euclidean domain
• Polynomial ring over a field

Weaker properties

• Multi-stage Euclidean domain
• Principal ideal domain
• Bezout domain
• Unique factorization domain

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:

• Multiplicatively monotone Euclidean norm

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).