Editing Euclidean domain (section)

==Definition==
Let {{mvar|R}} be an integral domain.  A '''Euclidean function''' on {{mvar|R}} is a [[function (mathematics)|function]] {{mvar|f}} from {{math|''R''&thinsp;\&hairsp;{0} }}to the non-negative integers satisfying the following fundamental division-with-remainder property:

*(EF1) If {{mvar|a}} and {{mvar|b}} are in {{mvar|R}} and {{mvar|b}} is nonzero, then there exist {{mvar|q}} and {{mvar|r}} in {{mvar|R}} such that {{math|''a'' {{=}} ''bq'' + ''r''}} and either {{math|1=''r'' = 0}} or {{math|''f''&thinsp;(''r'') < ''f''&thinsp;(''b'')}}.

A '''Euclidean domain''' is an integral domain which can be endowed with at least one Euclidean function. A particular Euclidean function {{mvar|f}} is ''not'' part of the definition of a Euclidean domain, as, in general, a Euclidean domain may admit many different Euclidean functions.

In this context, {{mvar|q}} and {{mvar|r}} are called respectively a ''quotient'' and a ''remainder'' of the ''division'' (or ''Euclidean division'') of {{mvar|a}} by {{mvar|b}}. In contrast with the case of [[integer]]s and [[polynomial]]s, the quotient is generally not uniquely defined, but when a quotient has been chosen, the remainder is uniquely defined. 

Most algebra texts require a Euclidean function to have the following additional property:

*(EF2) For all nonzero {{mvar|a}} and {{mvar|b}} in {{mvar|R}}, {{math|''f''&thinsp;(''a'') ≤ ''f''&thinsp;(''ab'')}}.

However, one can show that (EF1) alone suffices to define a Euclidean domain; if an integral domain {{mvar|R}} is endowed with a function {{mvar|g}} satisfying (EF1), then {{mvar|R}} can also be endowed with a function satisfying both (EF1) and (EF2) simultaneously. Indeed, for {{mvar|a}} in {{math|''R''&thinsp;\&hairsp;{0} }}, one can define {{math|''f''&thinsp;(''a'')}} as follows:<ref>{{Citation | last = Rogers | first = Kenneth | title = The Axioms for Euclidean Domains | journal = [[American Mathematical Monthly]] | volume = 78 | issue = 10 | pages = 1127–8 | year = 1971 | doi = 10.2307/2316324 | jstor = 2316324 | zbl=0227.13007 }}</ref>

:<math>f(a) = \min_{x \in R \setminus \{0\}} g(xa)</math>

In words, one may define {{math|''f''&thinsp;(''a'')}} to be the minimum value attained by {{mvar|g}} on the set of all non-zero elements of the principal ideal generated by {{mvar|a}}.

A Euclidean function {{mvar|f}} is '''multiplicative''' if {{math|''f''&thinsp;(''ab'') {{=}} ''f''&thinsp;(''a'')&thinsp;''f''&thinsp;(''b'')}} and {{math|''f''&thinsp;(''a'')}} is never zero. It follows that {{math|''f''&thinsp;(1) {{=}} 1}}. More generally, {{math|''f''&thinsp;(''a'') {{=}} 1}} if and only if {{mvar|a}} is a [[unit (ring theory)|unit]].

=== Notes on the definition ===

Many authors use other terms in place of "Euclidean function", such as "degree function", "valuation function", "gauge function" or "norm function".<ref name="DummitAlgebra">{{Cite book|title=Abstract Algebra|last1=Dummit|first1=David S.|last2=Foote|first2=Richard M.|publisher=Wiley|year=2004|isbn=9780471433347 |page=270}}</ref> Some authors also require the [[domain of a function|domain]] of the Euclidean function to be the entire ring {{mvar|R}};<ref name="DummitAlgebra"/> however, this does not essentially affect the definition, since (EF1) does not involve the value of {{math|''f''&thinsp;(0)}}. The definition is sometimes generalized by allowing the Euclidean function to take its values in any [[well-ordered set]]; this weakening does not affect the most important implications of the Euclidean property.

The property (EF1) can be restated as follows: for any principal ideal {{mvar|I}} of {{mvar|R}} with nonzero generator {{mvar|b}}, all nonzero classes of the [[quotient ring]] {{math|''R''/''I''}} have a representative {{mvar|r}} with {{math|''f''&thinsp;(''r'') < ''f''&thinsp;(''b'')}}. Since the possible values of {{mvar|f}} are well-ordered, this property can be established by showing that {{math|''f''&thinsp;(''r'') < ''f''&thinsp;(''b'')}} for any {{math|''r'' ∉ ''I''}} with minimal value of {{math|''f''&thinsp;(''r'')}} in its class. Note that, for a Euclidean function that is so established, there need not exist an effective method to determine {{mvar|q}} and {{mvar|r}} in (EF1).