Division in $mathbb Z[i]$ of $3+8i$ and $4+i$.












1












$begingroup$


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 ?






abstract-algebra divisibility







share|cite|improve this question









$endgroup$












  • $begingroup$
    Are you certain that there ought to be a single "correct" one?
    $endgroup$
    – Arthur
    Jan 4 at 16:38






  • 1




    $begingroup$
    In the ring of integers, we have $7=3cdot2+1,$ with $|1|<2$ and $7=4cdot2-1,$ with $|-1|<2.$
    $endgroup$
    – saulspatz
    Jan 4 at 16:59










  • $begingroup$
    Euclidean quotient and remainder are unique only in fields and univariate polynomial rings over fields, e.g. see here.
    $endgroup$
    – Bill Dubuque
    Jan 4 at 17:11










  • $begingroup$
    @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 $mathbb Z[isqrt{5}]$ ?
    $endgroup$
    – user623855
    Jan 4 at 20:12












  • $begingroup$
    @user623855 Examine closely the definition of a Euclidean domain.
    $endgroup$
    – Bill Dubuque
    Jan 4 at 20:23
















1












$begingroup$


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 ?






abstract-algebra divisibility







share|cite|improve this question









$endgroup$












  • $begingroup$
    Are you certain that there ought to be a single "correct" one?
    $endgroup$
    – Arthur
    Jan 4 at 16:38






  • 1




    $begingroup$
    In the ring of integers, we have $7=3cdot2+1,$ with $|1|<2$ and $7=4cdot2-1,$ with $|-1|<2.$
    $endgroup$
    – saulspatz
    Jan 4 at 16:59










  • $begingroup$
    Euclidean quotient and remainder are unique only in fields and univariate polynomial rings over fields, e.g. see here.
    $endgroup$
    – Bill Dubuque
    Jan 4 at 17:11










  • $begingroup$
    @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 $mathbb Z[isqrt{5}]$ ?
    $endgroup$
    – user623855
    Jan 4 at 20:12












  • $begingroup$
    @user623855 Examine closely the definition of a Euclidean domain.
    $endgroup$
    – Bill Dubuque
    Jan 4 at 20:23














1












1








1


1



$begingroup$


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 ?






abstract-algebra divisibility







share|cite|improve this question









$endgroup$




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 ?






abstract-algebra divisibility




abstract-algebra divisibility






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 4 at 16:20









user623855user623855

1407




1407












  • $begingroup$
    Are you certain that there ought to be a single "correct" one?
    $endgroup$
    – Arthur
    Jan 4 at 16:38






  • 1




    $begingroup$
    In the ring of integers, we have $7=3cdot2+1,$ with $|1|<2$ and $7=4cdot2-1,$ with $|-1|<2.$
    $endgroup$
    – saulspatz
    Jan 4 at 16:59










  • $begingroup$
    Euclidean quotient and remainder are unique only in fields and univariate polynomial rings over fields, e.g. see here.
    $endgroup$
    – Bill Dubuque
    Jan 4 at 17:11










  • $begingroup$
    @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 $mathbb Z[isqrt{5}]$ ?
    $endgroup$
    – user623855
    Jan 4 at 20:12












  • $begingroup$
    @user623855 Examine closely the definition of a Euclidean domain.
    $endgroup$
    – Bill Dubuque
    Jan 4 at 20:23


















  • $begingroup$
    Are you certain that there ought to be a single "correct" one?
    $endgroup$
    – Arthur
    Jan 4 at 16:38






  • 1




    $begingroup$
    In the ring of integers, we have $7=3cdot2+1,$ with $|1|<2$ and $7=4cdot2-1,$ with $|-1|<2.$
    $endgroup$
    – saulspatz
    Jan 4 at 16:59










  • $begingroup$
    Euclidean quotient and remainder are unique only in fields and univariate polynomial rings over fields, e.g. see here.
    $endgroup$
    – Bill Dubuque
    Jan 4 at 17:11










  • $begingroup$
    @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 $mathbb Z[isqrt{5}]$ ?
    $endgroup$
    – user623855
    Jan 4 at 20:12












  • $begingroup$
    @user623855 Examine closely the definition of a Euclidean domain.
    $endgroup$
    – Bill Dubuque
    Jan 4 at 20:23
















$begingroup$
Are you certain that there ought to be a single "correct" one?
$endgroup$
– Arthur
Jan 4 at 16:38




$begingroup$
Are you certain that there ought to be a single "correct" one?
$endgroup$
– Arthur
Jan 4 at 16:38




1




1




$begingroup$
In the ring of integers, we have $7=3cdot2+1,$ with $|1|<2$ and $7=4cdot2-1,$ with $|-1|<2.$
$endgroup$
– saulspatz
Jan 4 at 16:59




$begingroup$
In the ring of integers, we have $7=3cdot2+1,$ with $|1|<2$ and $7=4cdot2-1,$ with $|-1|<2.$
$endgroup$
– saulspatz
Jan 4 at 16:59












$begingroup$
Euclidean quotient and remainder are unique only in fields and univariate polynomial rings over fields, e.g. see here.
$endgroup$
– Bill Dubuque
Jan 4 at 17:11




$begingroup$
Euclidean quotient and remainder are unique only in fields and univariate polynomial rings over fields, e.g. see here.
$endgroup$
– Bill Dubuque
Jan 4 at 17:11












$begingroup$
@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 $mathbb Z[isqrt{5}]$ ?
$endgroup$
– user623855
Jan 4 at 20:12






$begingroup$
@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 $mathbb Z[isqrt{5}]$ ?
$endgroup$
– user623855
Jan 4 at 20:12














$begingroup$
@user623855 Examine closely the definition of a Euclidean domain.
$endgroup$
– Bill Dubuque
Jan 4 at 20:23




$begingroup$
@user623855 Examine closely the definition of a Euclidean domain.
$endgroup$
– Bill Dubuque
Jan 4 at 20:23










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