Editing Euclidean domain (section)

==Properties==
Let ''R'' be a domain and ''f'' a Euclidean function on ''R''.  Then:

* ''R'' is a [[principal ideal domain]] (PID). In fact, if ''I'' is a nonzero [[ideal (ring theory)|ideal]] of ''R'' then any element ''a'' of ''I''&nbsp;\&thinsp;{0} with minimal value (on that set) of ''f''(''a'') is a generator of ''I''.<ref>{{harvnb|Fraleigh|Katz|1967|p=377, Theorem 7.4}}</ref> As a consequence ''R'' is also a [[unique factorization domain]] and a [[Noetherian ring]]. With respect to general principal ideal domains, the existence of factorizations (i.e., that ''R'' is an [[atomic domain]]) is particularly easy to [[mathematical proof|prove]] in Euclidean domains: choosing a Euclidean function ''f'' satisfying (EF2), ''x'' cannot have any decomposition into more than ''f''(''x'') nonunit factors, so starting with ''x'' and repeatedly decomposing reducible factors is bound to produce a factorization into [[irreducible element]]s.
* Any element of ''R'' at which ''f'' takes its globally minimal value is invertible in ''R''.  If an ''f'' satisfying (EF2) is chosen, then the [[converse (logic)|converse]] also holds, and ''f'' takes its minimal value exactly at the invertible elements of ''R''.
*If Euclidean division is algorithmic, that is, if there is an [[algorithm]] for computing the quotient and the remainder, then an [[extended Euclidean algorithm]] can be defined exactly as in the case of integers.<ref>{{harvnb|Fraleigh|Katz|1967|p=380, Theorem 7.7}}</ref>
*If a Euclidean domain is not a field then it has a non-unit element ''a'' with the following property: any element ''x'' not divisible by ''a'' can be written as ''x'' = ''ay'' + ''u'' for some unit ''u'' and some element ''y''. This follows by taking ''a'' to be a non-unit with ''f''(''a'') as small as possible. This strange property can be used to show that some principal ideal domains are not Euclidean domains, as not all PIDs have this property.  For example, for ''d'' = −19, −43, −67, −163, the [[ring of integers]] of <math>\mathbf{Q}(\sqrt{d}\,)</math> is a PID which is {{em|not}} Euclidean (because it doesn't have this property), but the cases ''d'' = −1, −2, −3, −7, −11 {{em|are}} Euclidean.<ref>{{Citation
  | last = Motzkin | first = Theodore | author-link = Theodore Motzkin
  | title = The Euclidean algorithm
  | journal = [[Bulletin of the American Mathematical Society]]
  | volume = 55 | issue = 12 | pages = 1142–6 | year = 1949
  | url = http://projecteuclid.org/handle/euclid.bams/1183514381
  | doi = 10.1090/S0002-9904-1949-09344-8 | zbl=0035.30302
| doi-access = free}}</ref>

However, in many [[finite extension]]s of '''Q''' with [[trivial group|trivial]] [[Ideal class group|class group]], the ring of integers is Euclidean (not necessarily with respect to the absolute value of the field norm; see below).
Assuming the [[extended Riemann hypothesis]], if ''K'' is a finite [[field extension|extension]] of '''Q''' and the ring of integers of ''K'' is a PID with an infinite number of units, then the ring of integers is Euclidean.<ref>{{Citation
  | last = Weinberger | first = Peter J. | author-link = Peter J. Weinberger
  | title = On Euclidean rings of algebraic integers
  | journal = Proceedings of Symposia in Pure Mathematics
  | publisher = AMS
  | volume = 24 | pages = 321–332 | year = 1973
| doi = 10.1090/pspum/024/0337902 | isbn = 9780821814246 }}</ref>
In particular this applies to the case of [[totally real field|totally real]] [[quadratic field|quadratic number fields]] with trivial class group.
In addition (and without assuming ERH), if the field ''K'' is a [[Galois extension]] of '''Q''', has trivial class group and [[Dirichlet's unit theorem|unit rank]] strictly greater than three, then the ring of integers is Euclidean.<ref>{{Citation
  | last1 = Harper | first1 = Malcolm
  | last2 = Murty | first2 = M. Ram | author2-link = M. Ram Murty
  | title = Euclidean rings of algebraic integers
  | journal = Canadian Journal of Mathematics
  | volume = 56 | issue = 1 | pages = 71–76 | year = 2004
  | url = http://www.mast.queensu.ca/~murty/harper-murty.pdf
  | doi = 10.4153/CJM-2004-004-5
| citeseerx = 10.1.1.163.7917}}</ref>
An immediate [[corollary]] of this is that if the [[number field]] is Galois over '''Q''', its class group is trivial and the extension has [[degree of a field extension|degree]] greater than 8 then the ring of integers is necessarily Euclidean.
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== Euclidean domains according to Motzkin and Samuel ==
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