Editing Euclidean division (section)

==Division theorem==
{{anchor|Statement of the theorem}}
Euclidean division is based on the following result, which is sometimes called '''Euclid's division lemma'''.

Given two integers {{math|''a''}} and {{math|''b''}}, with {{math|''b'' ≠ 0}}, there exist [[Uniqueness quantification|unique]] integers {{math|''q''}} and {{math|''r''}} such that 
:{{math|1=''a'' = ''bq'' + ''r''}} 
and 
:{{math|0 ≤ ''r'' < {{!}}''b''{{!}}}}, 
where {{math|{{!}}''b''{{!}}}} denotes the [[absolute value]] of {{math|''b''}}.<ref>{{cite book |title=Elementary Number Theory |last=Burton |first=David M. |year=2010 |publisher=McGraw-Hill |isbn=978-0-07-338314-9 |pages=17–19}}</ref>

In the above theorem, each of the four integers has a name of its own: {{math|''a''}} is called the {{em|dividend}}, {{math|''b''}} is called the {{em|divisor}}, {{math|''q''}} is called the {{em|quotient}} and {{math|''r''}} is called the {{em|remainder}}.

The computation of the quotient and the remainder from the dividend and the divisor is called {{em|division}}, or in case of ambiguity, {{em|Euclidean division}}. The theorem is frequently referred to as the {{em|division algorithm}} (although it is a theorem and not an algorithm), because its proof as given below lends itself to a simple division algorithm for computing {{math|''q''}} and {{math|''r''}} (see the section [[#Proof|Proof]] for more).

Division is not defined in the case where {{math|1=''b'' = 0}}; see [[division by zero]].

For the remainder and the [[modulo operation]], there are conventions other than {{math|0 ≤ ''r'' < {{!}}''b''{{!}}}}, see {{slink||Other intervals for the remainder}}.

=== Generalization ===
{{main|Euclidean division of polynomials|Euclidean domain}}
Although originally restricted to integers, Euclidean division and the division theorem can be generalized to [[univariate polynomial|univariate polynomials]] over a [[field (mathematics)|field]] and to Euclidean domains. 

In the case of [[univariate polynomial]]s, the main difference is that the inequalities <math>0\le r<|b|</math> are replaced with
:<math>r = 0</math> or <math>\deg r < \deg b,</math>
where <math>\deg</math> denotes the [[polynomial degree]].

In the generalization to Euclidean domains, the inequality becomes
:<math>r = 0</math> or <math>f(r) < f(b),</math>
where <math>f</math> denote a specific function from the domain to the natural numbers called a "Euclidean function".

The uniqueness of the quotient and the remainder  remains true for polynomials, but it is false in general.