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单词 EuclideanValuation
释义

Euclidean valuation


Let D be an integral domain. A Euclidean valuation is a function from the nonzero elements of D to the nonnegative integers ν:D{0D}{x:x0} such that the following hold:

  • For any a,bD with b0D, there exist q,rD such that a=bq+r with ν(r)<ν(b) or r=0D.

  • For any a,bD{0D}, we have ν(a)ν(ab).

Euclidean valuations are important because they let us define greatest common divisorsMathworldPlanetmathPlanetmath and use Euclid’s algorithm. Some facts about Euclidean valuations include:

  • The minimal (http://planetmath.org/MinimalElement) value of ν is ν(1D). That is, ν(1D)ν(a) for any aD{0D}.

  • uD is a unit if and only if ν(u)=ν(1D).

  • For any aD{0D} and any unit u of D, we have ν(a)=ν(au).

These facts can be proven as follows:

  • If aD{0D}, then

    ν(1D)ν(1Da)=ν(a).
  • If uD is a unit, then let vD be its inversePlanetmathPlanetmath (http://planetmath.org/MultiplicativeInverse). Thus,

    ν(1D)ν(u)ν(uv)=ν(1D).

    Conversely, if ν(u)=ν(1D), then there exist q,rD with ν(r)<ν(u)=ν(1D) or r=0D such that

    1D=qu+r.

    Since ν(r)<ν(1D) is impossible, we must have r=0D. Hence, q is the inverse of u.

  • Let vD be the inverse of u. Then

    ν(a)ν(au)ν(auv)=ν(a).

Note that an integral domain is a Euclidean domain if and only if it has a Euclidean valuation.

Below are some examples of Euclidean domains and their Euclidean valuations:

  • Any field F is a Euclidean domain under the Euclidean valuation ν(a)=0 for all aF{0F}.

  • is a Euclidean domain with absolute valueMathworldPlanetmathPlanetmathPlanetmath acting as its Euclidean valuation.

  • If F is a field, then F[x], the ring of polynomials over F, is a Euclidean domain with degree acting as its Euclidean valuation: If n is a nonnegative integer and a0,,anF with an0F, then

    ν(j=0najxj)=n.

Due to the fact that the ring of polynomials over any field is always a Euclidean domain with degree acting as its Euclidean valuation, some refer to a Euclidean valuation as a degree function. This is done, for example, in Joseph J. Rotman’s .

TitleEuclidean valuation
Canonical nameEuclideanValuation
Date of creation2013-03-22 12:40:45
Last modified on2013-03-22 12:40:45
OwnerWkbj79 (1863)
Last modified byWkbj79 (1863)
Numerical id15
AuthorWkbj79 (1863)
Entry typeDefinition
Classificationmsc 13F07
Synonymdegree function
Related topicPID
Related topicUFD
Related topicRing
Related topicIntegralDomain
Related topicEuclideanRing
Related topicProofThatAnEuclideanDomainIsAPID
Related topicDedekindHasseValuation
Related topicEuclideanNumberField
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更新时间:2025/5/4 19:57:36