Euclidean domain - Revision history

Vipul: /* Definition */

2023-11-12T16:11:29Z

Definition

← Older revision		Revision as of 16:11, 12 November 2023
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	===Symbol-free definition===		===Symbol-free definition===

	An [[integral domain]] is said to be '''Euclidean''' if it admits a [[Euclidean norm]].		An [[integral domain]] is said to be '''Euclidean''' if it admits a [[defining ingredient::Euclidean norm]].

	===Definition with symbols===		===Definition with symbols===

Vipul: /* Relation with other properties */

2014-01-29T20:34:00Z

Relation with other properties

← Older revision		Revision as of 20:34, 29 January 2014
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	\| [[Stronger than::multi-stage Euclidean domain]] \|\| \|\| \|\| \|\| {{intermediate notions short\|multi-stage Euclidean domain\|Euclidean domain}}		\| [[Stronger than::multi-stage Euclidean domain]] \|\| \|\| \|\| \|\| {{intermediate notions short\|multi-stage Euclidean domain\|Euclidean domain}}
	\|-		\|-
	\| [[Stronger than::principal ideal domain]] \|\| [[integral domain]] that is a [[principal ideal ring]] \|\| \|\| \|\| {{intermediate notions short\|principal ideal domain\|Euclidean domain}}		\| [[Stronger than::principal ideal domain]] \|\| [[integral domain]] that is a [[principal ideal ring]] \|\| [[Euclidean implies PID]] \|\| [[PID not implies Euclidean]] \|\| {{intermediate notions short\|principal ideal domain\|Euclidean domain}}
	\|-		\|-
	\| [[Stronger than::Bezout domain]] \|\| [[integral domain]] in which every [[finitely generated ideal]] is [[principal ideal\|principal]] \|\| \|\| \|\| {{intermediate notions short\|Bezout domain\|Euclidean domain}}		\| [[Stronger than::Bezout domain]] \|\| [[integral domain]] in which every [[finitely generated ideal]] is [[principal ideal\|principal]] \|\| \|\| \|\| {{intermediate notions short\|Bezout domain\|Euclidean domain}}

Vipul: /* Weaker properties */

2013-07-18T04:23:06Z

Weaker properties

← Older revision		Revision as of 04:23, 18 July 2013
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	\|-		\|-
	\| [[Stronger than::Dedekind domain]] \|\| Noetherian, normal, one-dimensional domain \|\| \|\| \|\| {{intermediate notions short\|Dedekind domain\|Euclidean domain}}		\| [[Stronger than::Dedekind domain]] \|\| Noetherian, normal, one-dimensional domain \|\| \|\| \|\| {{intermediate notions short\|Dedekind domain\|Euclidean domain}}
			\|-
			\| [[Stronger than::Noetherian domain]] \|\| [[integral domain]] and every ideal is finitely generated \|\| \|\| \|\| {{intermediate notions short\|Noetherian domain\|Euclidean domain}}
			\|-
			\| [[Stronger than::Noetherian ring]] \|\| every ideal is finitely generated \|\| \|\| \|\| {{intermediate notions short\|Noetherian ring\|Euclidean domain}}
	\|}		\|}

Vipul: /* Relation with other properties */

2013-07-18T04:15:52Z

Relation with other properties

← Older revision		Revision as of 04:15, 18 July 2013
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	===Stronger properties===		===Stronger properties===

	* [[~~Stronger~~ than::~~Uniquely~~ Euclidean domain]]		{\| class="sortable" border="1"
	* [[~~Stronger~~ than::Polynomial ring over a field]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
			\| [[Weaker than::uniquely Euclidean domain]] \|\| there is a [[Euclidean norm]] for which Euclidean division is unique. \|\| \|\| \|\| {{intermediate notions short\|Euclidean domain\|uniquely Euclidean domain}}
			\|-
			\| [[Weaker than::Polynomial ring over a field]] \|\| it can be written as the [[polynomial ring]] <math>K[x]</math> for a [[field]] <math>K</math>. \|\| \|\| \|\| {{intermediate notions short\|Euclidean domain\|polynomial ring over a field}}
			\|}

	===Weaker properties===		===Weaker properties===

	* [[Stronger than::~~Multi~~-stage Euclidean domain]]		{\| class="sortable" border="1"
	* [[Stronger than::~~Principal~~ ideal domain]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* [[Stronger than::Bezout domain]]		\|-
	* [[Stronger than::~~Unique~~ factorization domain]]		\| [[Stronger than::multi-stage Euclidean domain]] \|\| \|\| \|\| \|\| {{intermediate notions short\|multi-stage Euclidean domain\|Euclidean domain}}
			\|-
			\| [[Stronger than::principal ideal domain]] \|\| [[integral domain]] that is a [[principal ideal ring]] \|\| \|\| \|\| {{intermediate notions short\|principal ideal domain\|Euclidean domain}}
			\|-
			\| [[Stronger than::Bezout domain]] \|\| [[integral domain]] in which every [[finitely generated ideal]] is [[principal ideal\|principal]] \|\| \|\| \|\| {{intermediate notions short\|Bezout domain\|Euclidean domain}}
			\|-
			\| [[Stronger than::unique factorization domain]] \|\| \|\| \|\| \|\| {{intermediate notions short\|unique factorization domain\|Euclidean domain}}
			\|-
			\| [[Stronger than::Dedekind domain]] \|\| Noetherian, normal, one-dimensional domain \|\| \|\| \|\| {{intermediate notions short\|Dedekind domain\|Euclidean domain}}
			\|}

	===Properties of Euclidean norms===		===Properties of Euclidean norms===
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	* [[Multiplicatively monotone Euclidean norm]]		* [[Multiplicatively monotone Euclidean norm]]

	==Metaproperties==		==Metaproperties==

Vipul: /* Examples */

2009-01-24T01:09:51Z

Examples

← Older revision		Revision as of 01:09, 24 January 2009
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	===Standard examples===		===Standard examples===

	* The [[ring of integers]] <math>\mathbb{Z}</math> is a Euclidean domain with Euclidean norm defined by the absolute value. {{proofat\|[[Ring of integers is Euclidean with norm equal to absolute value]]}}		* The [[ring of rational integers]] <math>\mathbb{Z}</math> is a Euclidean domain with Euclidean norm defined by the absolute value. {{proofat\|[[Ring of integers is Euclidean with norm equal to absolute value]]}}
	* The [[polynomial ring over a field]] <math>k[x]</math> 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. {{proofat\|[[Polynomial ring over a field is uniquely Euclidean with norm equal to degree]]}}		* The [[polynomial ring over a field]] <math>k[x]</math> 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. {{proofat\|[[Polynomial ring over a field is uniquely Euclidean with norm equal to degree]]}}

Vipul at 02:30, 23 January 2009

2009-01-23T02:30:19Z

← Older revision		Revision as of 02:30, 23 January 2009
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	We call <math>a</math> the ''dividend'', <math>b</math> the ''divisor'', <math>q</math> the ''quotient'' and <math>r</math> the ''remainder''.		We call <math>a</math> the ''dividend'', <math>b</math> the ''divisor'', <math>q</math> the ''quotient'' and <math>r</math> the ''remainder''.

	Such a function <math>N</math> is called a ''Euclidean norm'' on <math>R</math>.		Such a function <math>N</math> is called a [[Euclidean norm]] on <math>R</math>.

	The definition of Euclidean domain does not require that <math>q</math> and <math>r</math> be uniquely determined from <math>a</math> and <math>b</math>. If <math>q</math> and <math>r</math> are uniquely determined from <math>a</math> and <math>b</math>, the integral domain is termed a [[uniquely Euclidean domain]].		===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 <math>q</math> and <math>r</math> be uniquely determined from <math>a</math> and <math>b</math>. If <math>q</math> and <math>r</math> are uniquely determined from <math>a</math> and <math>b</math>, the integral domain is termed a [[uniquely Euclidean domain]].

			==Examples==

			===Standard examples===

			* The [[ring of integers]] <math>\mathbb{Z}</math> is a Euclidean domain with Euclidean norm defined by the absolute value. {{proofat\|[[Ring of integers is Euclidean with norm equal to absolute value]]}}
			* The [[polynomial ring over a field]] <math>k[x]</math> 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. {{proofat\|[[Polynomial ring over a field is uniquely Euclidean with norm equal to degree]]}}

			===Other examples===

			* The [[ring of Gaussian integers]] <math>\mathbb{Z}[i]</math> is a Euclidean domain with Euclidean norm equal to the norm in the sense of a quadratic integer ring. {{proofat\|[[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 <math>n</math> for which the coefficient of <math>x^n</math> 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==		==Relation with other properties==
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	===Stronger properties===		===Stronger properties===

	* [[Uniquely Euclidean domain]]		* [[Stronger than::Uniquely Euclidean domain]]
	* [[Polynomial ring over a field]]		* [[Stronger than::Polynomial ring over a field]]

	===Weaker properties===		===Weaker properties===

	* [[Multi-stage Euclidean domain]]		* [[Stronger than::Multi-stage Euclidean domain]]
	* [[Principal ideal domain]]		* [[Stronger than::Principal ideal domain]]
	* [[Bezout domain]]		* [[Stronger than::Bezout domain]]
	* [[Unique factorization domain]]		* [[Stronger than::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:

			[[:Category:Properties of Euclidean norms]]

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

			* [[Multiplicatively monotone Euclidean norm]]
	==Metaproperties==		==Metaproperties==

Vipul: /* Definition */

2009-01-22T16:53:26Z

Definition

← Older revision		Revision as of 16:53, 22 January 2009
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	We call <math>a</math> the ''dividend'', <math>b</math> the ''divisor'', <math>q</math> the ''quotient'' and <math>r</math> the ''remainder''.		We call <math>a</math> the ''dividend'', <math>b</math> the ''divisor'', <math>q</math> the ''quotient'' and <math>r</math> the ''remainder''.

			Such a function <math>N</math> is called a ''Euclidean norm'' on <math>R</math>.

	The definition of Euclidean domain does not require that <math>q</math> and <math>r</math> be uniquely determined from <math>a</math> and <math>b</math>. If <math>q</math> and <math>r</math> are uniquely determined from <math>a</math> and <math>b</math>, the integral domain is termed a [[uniquely Euclidean domain]].		The definition of Euclidean domain does not require that <math>q</math> and <math>r</math> be uniquely determined from <math>a</math> and <math>b</math>. If <math>q</math> and <math>r</math> are uniquely determined from <math>a</math> and <math>b</math>, the integral domain is termed a [[uniquely Euclidean domain]].

Vipul: 7 revisions

2008-05-12T16:20:04Z

7 revisions

← Older revision	Revision as of 16:20, 12 May 2008
(No difference)

Vipul at 22:52, 16 December 2007

2007-12-16T22:52:08Z

← Older revision		Revision as of 22:52, 16 December 2007
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	* [[Uniquely Euclidean domain]]		* [[Uniquely Euclidean domain]]
			* [[Polynomial ring over a field]]

	===Weaker properties===		===Weaker properties===
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	* [[Bezout domain]]		* [[Bezout domain]]
	* [[Unique factorization domain]]		* [[Unique factorization domain]]

			==Metaproperties==

			{{not poly-closed curing property}}

			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).

Vipul at 20:37, 23 November 2007

2007-11-23T20:37:49Z

← Older revision		Revision as of 20:37, 23 November 2007
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	We call <math>a</math> the ''dividend'', <math>b</math> the ''divisor'', <math>q</math> the ''quotient'' and <math>r</math> the ''remainder''.		We call <math>a</math> the ''dividend'', <math>b</math> the ''divisor'', <math>q</math> the ''quotient'' and <math>r</math> the ''remainder''.

	The definition of Euclidean domain does not require that <math>q</math> and <math>r</math> be uniquely determined from <math>a</math> and <math>b</math>. If <math>q</math> and <math>r</math>a are uniquely determined from <math>a</math> and <math>b</math>, the integral domain is termed a [[uniquely Euclidean domain]].		The definition of Euclidean domain does not require that <math>q</math> and <math>r</math> be uniquely determined from <math>a</math> and <math>b</math>. If <math>q</math> and <math>r</math> are uniquely determined from <math>a</math> and <math>b</math>, the integral domain is termed a [[uniquely Euclidean domain]].

	==Relation with other properties==		==Relation with other properties==