Euclidean valuation
Let be an integral domain. A Euclidean valuation is a function from the nonzero elements of to the nonnegative integers such that the following hold:
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For any with , there exist such that with or .
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For any , we have .
Euclidean valuations are important because they let us define greatest common divisors and use Euclid’s algorithm. Some facts about Euclidean valuations include:
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The minimal (http://planetmath.org/MinimalElement) value of is . That is, for any .
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is a unit if and only if .
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For any and any unit of , we have .
These facts can be proven as follows:
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If , then
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If is a unit, then let be its inverse (http://planetmath.org/MultiplicativeInverse). Thus,
Conversely, if , then there exist with or such that
Since is impossible, we must have . Hence, is the inverse of .
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Let be the inverse of . Then
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:
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Any field is a Euclidean domain under the Euclidean valuation for all .
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is a Euclidean domain with absolute value acting as its Euclidean valuation.
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If is a field, then , the ring of polynomials over , is a Euclidean domain with degree acting as its Euclidean valuation: If is a nonnegative integer and with , then
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 .
Title | Euclidean valuation |
Canonical name | EuclideanValuation |
Date of creation | 2013-03-22 12:40:45 |
Last modified on | 2013-03-22 12:40:45 |
Owner | Wkbj79 (1863) |
Last modified by | Wkbj79 (1863) |
Numerical id | 15 |
Author | Wkbj79 (1863) |
Entry type | Definition |
Classification | msc 13F07 |
Synonym | degree function |
Related topic | PID |
Related topic | UFD |
Related topic | Ring |
Related topic | IntegralDomain |
Related topic | EuclideanRing |
Related topic | ProofThatAnEuclideanDomainIsAPID |
Related topic | DedekindHasseValuation |
Related topic | EuclideanNumberField |