Editing Euclidean domain (section)

==Examples==
Examples of Euclidean domains include:

*Any field. Define {{math|''f''&thinsp;(''x'') {{=}} 1}} for all nonzero {{mvar|x}}.
*{{math|'''Z'''}}, the ring of integers. Define {{math|''f''&thinsp;(''n'') {{=}} {{!}}''n''{{!}}}}, the [[absolute value]] of {{mvar|n}}.<ref>{{harvnb|Fraleigh|Katz|1967|p=377, Example 1}}</ref>
*{{math|'''Z'''[{{hairsp|''i''}}]}}, the ring of [[Gaussian integer]]s. Define {{math|''f''&thinsp;(''a'' + ''bi'') {{=}} ''a''{{sup|2}} + ''b''{{sup|2}}}}, the [[field norm|norm]] of the Gaussian integer {{math|''a'' + ''bi''}}.
* {{math|'''Z'''[ω]}} (where {{math|ω}} is a [[Root of unity#General definition|primitive]] (non-[[real number|real]]) [[cube root of unity]]), the ring of [[Eisenstein integer]]s. Define {{math|''f''&thinsp;(''a'' + ''b''ω) {{=}} ''a''{{sup|2}} − ''ab'' + ''b''{{sup|2}}}}, the norm of the Eisenstein integer {{math|''a'' + ''b''ω}}.
*{{math|''K''[''X'']}}, the [[polynomial ring|ring of polynomials]] over a [[field (mathematics)|field]] {{mvar|K}}. For each nonzero polynomial {{mvar|P}}, define {{math|''f''&thinsp;(''P'')}} to be the [[degree of a polynomial|degree]] of {{mvar|P}}.<ref>{{harvnb|Fraleigh|Katz|1967|p=377, Example 2}}</ref>
*{{math|''K''{{brackets|''X''}}}}, the ring of [[formal power series]] over the field {{mvar|K}}. For each nonzero [[power series]] {{mvar|P}}, define {{math|''f''&thinsp;(''P'')}} as the [[order (power series)|order]] of {{mvar|P}}, that is the degree of the smallest power of {{mvar|X}} occurring in {{mvar|P}}. In particular, for two nonzero power series {{mvar|P}} and {{mvar|Q}}, {{math|''f''&thinsp;(''P'') ≤ ''f''&thinsp;(''Q'')}} if and only if {{mvar|P}} [[Formal power series#Dividing series|divides]] {{mvar|Q}}.
*Any [[discrete valuation ring]]. Define {{math|''f''&thinsp;(''x'')}} to be the highest power of the [[maximal ideal]] {{mvar|M}} containing {{mvar|x}}. Equivalently, let {{mvar|g}} be a generator of {{mvar|M}}, and {{mvar|v}} be the unique integer such that {{mvar|g{{hairsp}}{{sup|v}}}} is an [[associated elements|associate]] of {{mvar|x}}, then define {{math|''f''&thinsp;(''x'') {{=}} ''v''}}. The previous example {{math|''K''{{brackets|''X''}}}} is a special case of this.
*A [[Dedekind domain]] with finitely many [[zero ideal|nonzero]] [[prime ideal]]s {{math|''P''{{sub|1}}, ..., ''P{{sub|n}}''}}.  Define <math>f(x) = \sum_{i=1}^n v_i(x)</math>, where {{mvar|v{{sub|i}}}} is the [[discrete valuation]] corresponding to the ideal {{mvar|P{{sub|i}}}}.<ref>{{Cite journal|last=Samuel|first=Pierre|date=1 October 1971|title=About Euclidean rings|journal=Journal of Algebra|volume=19|issue=2|pages=282–301 (p. 285)|doi=10.1016/0021-8693(71)90110-4|issn=0021-8693|doi-access=free}}</ref>

Examples of domains that are ''not'' Euclidean domains include:
* Every domain that is not a [[principal ideal domain]], such as the ring of polynomials in at least two indeterminates over a field, or the ring of univariate polynomials with integer [[coefficient]]s, or the number ring {{math|'''Z'''[{{hairsp|{{sqrt|&minus;5}}}}]}}.
* The [[ring of integers]] of {{math|'''Q'''({{hairsp|{{sqrt|−19}}}})}}, consisting of the numbers {{math|{{sfrac|''a'' + ''b''{{sqrt|&minus;19}}|2}}}} where {{mvar|a}} and {{mvar|b}} are integers and both even or both odd. It is a principal ideal domain that is not Euclidean.This was proved by [[Theodore Motzkin]] and was the first case known.<ref>{{Cite journal |last=Motzkin |first=Th |date=December 1949 |title=The Euclidean algorithm |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-55/issue-12/The-Euclidean-algorithm/bams/1183514381.full |journal=Bulletin of the American Mathematical Society |volume=55 |issue=12 |pages=1142–1146 |doi=10.1090/S0002-9904-1949-09344-8 |issn=0002-9904|doi-access=free }}</ref>
* The ring {{math|1=''A'' = '''R'''[''X'', ''Y'']/(''X''<sup>&thinsp;2</sup> + ''Y''<sup>&thinsp;2</sup> + 1)}} is also a principal ideal domain<ref>
{{cite book|last=Pierre|first=Samuel|url=http://www.math.tifr.res.in/~publ/ln/tifr30.pdf|title=Lectures on Unique Factorization Domains|date=1964|publisher=Tata Institute of Fundamental Research|isbn= |pages=27–28|author-link=}}
</ref> that is not Euclidean. To see that it is not a Euclidean domain, it suffices to show that for every non-zero prime <math>p\in A</math>, the map <math>A^\times\to(A/p)^\times</math> induced by the quotient map <math>A\to A/p</math> is not [[surjective]].<ref>
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