Division in $mathbb Z[i]$ of $3+8i$ and $4+i$.
1
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
asked Jan 4 at 16:20
user623855user623855
807
|
show 2 more comments
1
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
asked Jan 4 at 16:20
user623855user623855
807
Are you certain that there ought to be a single "correct" one?
– Arthur
Jan 4 at 16:38
1
In the ring of integers, we have $7=3cdot2+1,$ with $|1|<2$ and $7=4cdot2-1,$ with $|-1|<2.$
– saulspatz
Jan 4 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 4 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 $mathbb Z[isqrt{5}]$ ?
– user623855
Jan 4 at 20:12
@user623855 Examine closely the definition of a Euclidean domain.
– Bill Dubuque
Jan 4 at 20:23
|
show 2 more comments
1
1
1
1
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
asked Jan 4 at 16:20
user623855user623855
807
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
asked Jan 4 at 16:20
user623855user623855
807
asked Jan 4 at 16:20
user623855user623855
807
asked Jan 4 at 16:20
user623855user623855
807
asked Jan 4 at 16:20
user623855user623855
807
asked Jan 4 at 16:20
user623855user623855
807
807
Are you certain that there ought to be a single "correct" one?
– Arthur
Jan 4 at 16:38
1
In the ring of integers, we have $7=3cdot2+1,$ with $|1|<2$ and $7=4cdot2-1,$ with $|-1|<2.$
– saulspatz
Jan 4 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 4 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 $mathbb Z[isqrt{5}]$ ?
– user623855
Jan 4 at 20:12
@user623855 Examine closely the definition of a Euclidean domain.
– Bill Dubuque
Jan 4 at 20:23
|
show 2 more comments
Are you certain that there ought to be a single "correct" one?
– Arthur
Jan 4 at 16:38
1
In the ring of integers, we have $7=3cdot2+1,$ with $|1|<2$ and $7=4cdot2-1,$ with $|-1|<2.$
– saulspatz
Jan 4 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 4 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 $mathbb Z[isqrt{5}]$ ?
– user623855
Jan 4 at 20:12
@user623855 Examine closely the definition of a Euclidean domain.
– Bill Dubuque
Jan 4 at 20:23
Are you certain that there ought to be a single "correct" one?
– Arthur
Jan 4 at 16:38
Are you certain that there ought to be a single "correct" one?
– Arthur
Jan 4 at 16:38
1
1
In the ring of integers, we have $7=3cdot2+1,$ with $|1|<2$ and $7=4cdot2-1,$ with $|-1|<2.$
– saulspatz
Jan 4 at 16:59
In the ring of integers, we have $7=3cdot2+1,$ with $|1|<2$ and $7=4cdot2-1,$ with $|-1|<2.$
– saulspatz
Jan 4 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 4 at 17:11
Euclidean quotient and remainder are unique only in fields and univariate polynomial rings over fields, e.g. see here.
– Bill Dubuque
Jan 4 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 $mathbb Z[isqrt{5}]$ ?
– user623855
Jan 4 at 20:12
@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 $mathbb Z[isqrt{5}]$ ?
– user623855
Jan 4 at 20:12
@user623855 Examine closely the definition of a Euclidean domain.
– Bill Dubuque
Jan 4 at 20:23
@user623855 Examine closely the definition of a Euclidean domain.
– Bill Dubuque
Jan 4 at 20:23
|
show 2 more comments
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Are you certain that there ought to be a single "correct" one?
– Arthur
Jan 4 at 16:38
1
In the ring of integers, we have $7=3cdot2+1,$ with $|1|<2$ and $7=4cdot2-1,$ with $|-1|<2.$
– saulspatz
Jan 4 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 4 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 $mathbb Z[isqrt{5}]$ ?
– user623855
Jan 4 at 20:12
@user623855 Examine closely the definition of a Euclidean domain.
– Bill Dubuque
Jan 4 at 20:23