Editing Euclidean domain (section)

== Norm-Euclidean fields ==

[[Algebraic number field]]s ''K'' come with a canonical norm function on them: the absolute value of the [[field norm]] ''N'' that takes an [[algebraic element]] ''α'' to the product of all the [[Conjugate element (field theory)|conjugates]] of ''α''.  This norm maps the [[ring of integers]] of a number field ''K'', say ''O''<sub>''K''</sub>, to the nonnegative [[Integer|rational integers]], so it is a candidate to be a Euclidean norm on this ring.  If this norm satisfies the axioms of a Euclidean function then the number field ''K'' is called ''norm-Euclidean'' or simply ''Euclidean''.<ref name="RibAlgNum">{{cite book | title=Algebraic Numbers | publisher=Wiley-Interscience | author=Ribenboim, Paulo | year=1972 | isbn=978-0-471-71804-8}}</ref><ref name="HardyWright">{{cite book |first1=G.H. |last1=Hardy |first2=E.M. |last2=Wright |first3=Joseph |last3=Silverman |first4=Andrew |last4=Wiles |title=An Introduction to the Theory of Numbers |url=https://books.google.com/books?id=P6uTBqOa3T4C&pg=PP1 |date=2008 |publisher=Oxford University Press |edition=6th |isbn=978-0-19-921986-5 }}</ref>  Strictly speaking it is the ring of integers that is Euclidean since fields are trivially Euclidean domains, but the terminology is standard.

If a field is not norm-Euclidean then that does not mean the ring of integers is not Euclidean, just that the field norm does not satisfy the axioms of a Euclidean function. In fact, the rings of integers of number fields may be divided in several classes:
*Those that are not [[principal ideal domain|principal]] and therefore not Euclidean, such as the integers of <math>\mathbf{Q}(\sqrt{-5}\,)</math>
*Those that are principal and not Euclidean, such as the integers of <math>\mathbf{Q}(\sqrt{-19}\,)</math>
*Those that are Euclidean and not norm-Euclidean, such as the integers of <math>\mathbf{Q}(\sqrt{69}\,)</math><ref>{{cite journal | last=Clark | first=David A. | title=A quadratic field which is Euclidean but not norm-Euclidean | journal=[[Manuscripta Mathematica]] | volume=83 | number=3–4 | pages=327–330 | year=1994 | doi = 10.1007/BF02567617 | zbl=0817.11047 | citeseerx=10.1.1.360.6129 }}</ref>
*Those that are norm-Euclidean, such as [[Gaussian integer]]s (integers of <math>\mathbf{Q}(\sqrt{-1}\,)</math>)

The norm-Euclidean [[quadratic field]]s have been fully classified; they are <math>\mathbf{Q}(\sqrt{d}\,)</math> where <math>d</math> takes the values
:−11, −7, −3, −2, −1, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, 73 {{OEIS|id=A048981}}.<ref>{{cite book | last = LeVeque | first = William J. | author-link = William J. LeVeque | title = Topics in Number Theory|volume=I and II | publisher = Dover | year = 2002 | orig-year = 1956 | isbn = 978-0-486-42539-9 | zbl = 1009.11001 | pages = [https://archive.org/details/topicsinnumberth0000leve/page/ II:57,81] | url = https://archive.org/details/topicsinnumberth0000leve/page/ }}</ref>

Every Euclidean imaginary quadratic field is norm-Euclidean and is one of the five first fields in the preceding list.
<!--
== Euclidean rings with zero-divisors ==
== ''k''-stage Euclidean domains ==
== Euclidean ideal classes ==
-->