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Let $a=3+8i$ and $b=4+i$. We have that $$\frac{a}{b}=\frac{20}{17}+\frac{29}{17}i.$$

So $a=b(1+2i)+1-i$ or $a=b(1+i)+3i$. In both case we have that $N(b)=17$ and $N(1-i)=2<N(b)$ and $N(3i)=9<N(b)$. So which euclidienne division is the right one ?

user623855
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  • Are you certain that there ought to be a single "correct" one? – Arthur Jan 04 '19 at 16:38
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    In the ring of integers, we have $7=3\cdot2+1,$ with $|1|<2$ and $7=4\cdot2-1,$ with $|-1|<2.$ – saulspatz Jan 04 '19 at 16:59
  • Euclidean quotient and remainder are unique only in fields and univariate polynomial rings over fields, e.g. see here. – Bill Dubuque Jan 04 '19 at 17:11
  • @Arthur: If it's not unique (modulo a certain condition), why do we consider euclidien rings ? Indeed, in any ring I can write the couple $(a,b)$ as $a=qb+r$... So if there is no either a unicity or a sort of unicity, why do we consider such a writting in $\mathbb Z[i]$ and not in $\mathbb Z[i\sqrt{5}]$ ? – user623855 Jan 04 '19 at 20:12
  • @user623855 Examine closely the definition of a Euclidean domain. – Bill Dubuque Jan 04 '19 at 20:23
  • @BillDubuque: Thank you for your contribution. My definition is : A domain $R$ is euclidien if there is an algebraic norm s.t. for all $a,b\in R$ there is $q,r\in R$ s.t. $a=bq+r$ where $r=0$ or $N(r)<N(b)$. But maybe there is a subtlety that I don't see. – user623855 Jan 04 '19 at 20:33
  • @user623855 What is an "algebraic norm"? Try applying that to $,\Bbb Z,$ with $,N(x) = |x|,$ and you'll observe the nonuniqueness exemplified in saulspatz's comment above. – Bill Dubuque Jan 04 '19 at 21:10

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