If $fcolonmathbb{Z}to F$ is an onto morphism and $F$ is a field, then $mathbb{Z}_pcong F$ where $p$ is a...
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Let $F$ a field ed $fcolonmathbb{Z}to F$ an onto morphism. We know that $ker f$ is an ideal of $mathbb{Z}$, then $ker f=(n)=nmathbb{Z}$ for same $ninmathbb{Z}$. For the first isomorphism theorem we have that $$mathbb{Z}_n:=mathbb{Z}/nmathbb{Z}=mathbb{Z}/ker fcong F.$$
Now, if $n=0$, the canonical projection is $picolonmathbb{Z}to mathbb{Z}$, but the only non-zero morphism from $mathbb{Z}$ to $mathbb{Z}$ is $id_{mathbb{Z}}$ which is, in particular, injective. Since $ker f=nmathbb{Z}$, then $tilde{f}colonmathbb{Z}to F$ is injective, but for the first isomorphism theorem $f=tilde{f}circpi$, then $f$ is injective, moreover, for hypotesis, $f$ is onto, then $f$ is an isomorphism. But this is absurd, because $mathbb{Z}$ is not a field. Therefore $n$ can not be $0$.
On the other hand $mathbb{Z}_ncong F$, then $mathbb{Z}_n$ is a field, then $n$ must be prime.
Correct?
Thanks!
abstract-algebra proof-verification proof-explanation
asked Nov 20 at 9:04
Jack J.
4101317
|
show 2 more comments
up vote
0
down vote
favorite
Let $F$ a field ed $fcolonmathbb{Z}to F$ an onto morphism. We know that $ker f$ is an ideal of $mathbb{Z}$, then $ker f=(n)=nmathbb{Z}$ for same $ninmathbb{Z}$. For the first isomorphism theorem we have that $$mathbb{Z}_n:=mathbb{Z}/nmathbb{Z}=mathbb{Z}/ker fcong F.$$
Now, if $n=0$, the canonical projection is $picolonmathbb{Z}to mathbb{Z}$, but the only non-zero morphism from $mathbb{Z}$ to $mathbb{Z}$ is $id_{mathbb{Z}}$ which is, in particular, injective. Since $ker f=nmathbb{Z}$, then $tilde{f}colonmathbb{Z}to F$ is injective, but for the first isomorphism theorem $f=tilde{f}circpi$, then $f$ is injective, moreover, for hypotesis, $f$ is onto, then $f$ is an isomorphism. But this is absurd, because $mathbb{Z}$ is not a field. Therefore $n$ can not be $0$.
On the other hand $mathbb{Z}_ncong F$, then $mathbb{Z}_n$ is a field, then $n$ must be prime.
Correct?
Thanks!
abstract-algebra proof-verification proof-explanation
asked Nov 20 at 9:04
Jack J.
4101317
1
I don't understand your claim. $mathbb{Z}$ is not isomorphic to a field because it is not a field: $2$ has no multiplicative inverse. And there is an onto homomorphism from $mathbb{Z}$ to a field, take the quotient map $mathbb{Z}tomathbb{Z}/pmathbb{Z}$ for prime $p$. But obviously not every field is a homomorphic image of $mathbb{Z}$ (e.g. reals).
– freakish
Nov 20 at 9:11
Sorry but I have not explained well, now I correct my question.
– Jack J.
Nov 20 at 9:18
The kernel cannot be zero, since this would imply that $2$, for exemple, has multiplicative inverse (since its image in F does). Now, by the isomorphism theorem you have $Z/nZ cong F$. Thus, $Z/nZ$ is a field. Now prove that $n$ must be prime.
– math.h
Nov 20 at 10:17
@math.h I would like to know if the way I have shown that $n$ can not be zero is correct
– Jack J.
Nov 20 at 10:23
1
Yes, it is. But I think you are "talking too much" (no offense, really) for just a simple thing. The morphism $Zrightarrow F$ cannot be injective (otherwise the morphism would be bijective), since $F$ is a field. You seem to know that $Z$ can't be isomorphic to a field, so you gotta accept this argument. As you said, the kernel should be of the form: $nZ$. As it is not zero (the kernel), then $n$ cannot be zero.
– math.h
Nov 20 at 10:35
|
show 2 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $F$ a field ed $fcolonmathbb{Z}to F$ an onto morphism. We know that $ker f$ is an ideal of $mathbb{Z}$, then $ker f=(n)=nmathbb{Z}$ for same $ninmathbb{Z}$. For the first isomorphism theorem we have that $$mathbb{Z}_n:=mathbb{Z}/nmathbb{Z}=mathbb{Z}/ker fcong F.$$
Now, if $n=0$, the canonical projection is $picolonmathbb{Z}to mathbb{Z}$, but the only non-zero morphism from $mathbb{Z}$ to $mathbb{Z}$ is $id_{mathbb{Z}}$ which is, in particular, injective. Since $ker f=nmathbb{Z}$, then $tilde{f}colonmathbb{Z}to F$ is injective, but for the first isomorphism theorem $f=tilde{f}circpi$, then $f$ is injective, moreover, for hypotesis, $f$ is onto, then $f$ is an isomorphism. But this is absurd, because $mathbb{Z}$ is not a field. Therefore $n$ can not be $0$.
On the other hand $mathbb{Z}_ncong F$, then $mathbb{Z}_n$ is a field, then $n$ must be prime.
Correct?
Thanks!
abstract-algebra proof-verification proof-explanation
asked Nov 20 at 9:04
Jack J.
4101317
Let $F$ a field ed $fcolonmathbb{Z}to F$ an onto morphism. We know that $ker f$ is an ideal of $mathbb{Z}$, then $ker f=(n)=nmathbb{Z}$ for same $ninmathbb{Z}$. For the first isomorphism theorem we have that $$mathbb{Z}_n:=mathbb{Z}/nmathbb{Z}=mathbb{Z}/ker fcong F.$$
Now, if $n=0$, the canonical projection is $picolonmathbb{Z}to mathbb{Z}$, but the only non-zero morphism from $mathbb{Z}$ to $mathbb{Z}$ is $id_{mathbb{Z}}$ which is, in particular, injective. Since $ker f=nmathbb{Z}$, then $tilde{f}colonmathbb{Z}to F$ is injective, but for the first isomorphism theorem $f=tilde{f}circpi$, then $f$ is injective, moreover, for hypotesis, $f$ is onto, then $f$ is an isomorphism. But this is absurd, because $mathbb{Z}$ is not a field. Therefore $n$ can not be $0$.
On the other hand $mathbb{Z}_ncong F$, then $mathbb{Z}_n$ is a field, then $n$ must be prime.
Correct?
Thanks!
abstract-algebra proof-verification proof-explanation
abstract-algebra proof-verification proof-explanation
asked Nov 20 at 9:04
Jack J.
4101317
asked Nov 20 at 9:04
Jack J.
4101317
edited Nov 20 at 9:21
asked Nov 20 at 9:04
Jack J.
4101317
asked Nov 20 at 9:04
Jack J.
4101317
asked Nov 20 at 9:04
Jack J.
4101317
4101317
1
I don't understand your claim. $mathbb{Z}$ is not isomorphic to a field because it is not a field: $2$ has no multiplicative inverse. And there is an onto homomorphism from $mathbb{Z}$ to a field, take the quotient map $mathbb{Z}tomathbb{Z}/pmathbb{Z}$ for prime $p$. But obviously not every field is a homomorphic image of $mathbb{Z}$ (e.g. reals).
– freakish
Nov 20 at 9:11
Sorry but I have not explained well, now I correct my question.
– Jack J.
Nov 20 at 9:18
The kernel cannot be zero, since this would imply that $2$, for exemple, has multiplicative inverse (since its image in F does). Now, by the isomorphism theorem you have $Z/nZ cong F$. Thus, $Z/nZ$ is a field. Now prove that $n$ must be prime.
– math.h
Nov 20 at 10:17
@math.h I would like to know if the way I have shown that $n$ can not be zero is correct
– Jack J.
Nov 20 at 10:23
1
Yes, it is. But I think you are "talking too much" (no offense, really) for just a simple thing. The morphism $Zrightarrow F$ cannot be injective (otherwise the morphism would be bijective), since $F$ is a field. You seem to know that $Z$ can't be isomorphic to a field, so you gotta accept this argument. As you said, the kernel should be of the form: $nZ$. As it is not zero (the kernel), then $n$ cannot be zero.
– math.h
Nov 20 at 10:35
|
show 2 more comments
1
I don't understand your claim. $mathbb{Z}$ is not isomorphic to a field because it is not a field: $2$ has no multiplicative inverse. And there is an onto homomorphism from $mathbb{Z}$ to a field, take the quotient map $mathbb{Z}tomathbb{Z}/pmathbb{Z}$ for prime $p$. But obviously not every field is a homomorphic image of $mathbb{Z}$ (e.g. reals).
– freakish
Nov 20 at 9:11
Sorry but I have not explained well, now I correct my question.
– Jack J.
Nov 20 at 9:18
The kernel cannot be zero, since this would imply that $2$, for exemple, has multiplicative inverse (since its image in F does). Now, by the isomorphism theorem you have $Z/nZ cong F$. Thus, $Z/nZ$ is a field. Now prove that $n$ must be prime.
– math.h
Nov 20 at 10:17
@math.h I would like to know if the way I have shown that $n$ can not be zero is correct
– Jack J.
Nov 20 at 10:23
1
Yes, it is. But I think you are "talking too much" (no offense, really) for just a simple thing. The morphism $Zrightarrow F$ cannot be injective (otherwise the morphism would be bijective), since $F$ is a field. You seem to know that $Z$ can't be isomorphic to a field, so you gotta accept this argument. As you said, the kernel should be of the form: $nZ$. As it is not zero (the kernel), then $n$ cannot be zero.
– math.h
Nov 20 at 10:35
1
1
I don't understand your claim. $mathbb{Z}$ is not isomorphic to a field because it is not a field: $2$ has no multiplicative inverse. And there is an onto homomorphism from $mathbb{Z}$ to a field, take the quotient map $mathbb{Z}tomathbb{Z}/pmathbb{Z}$ for prime $p$. But obviously not every field is a homomorphic image of $mathbb{Z}$ (e.g. reals).
– freakish
Nov 20 at 9:11
I don't understand your claim. $mathbb{Z}$ is not isomorphic to a field because it is not a field: $2$ has no multiplicative inverse. And there is an onto homomorphism from $mathbb{Z}$ to a field, take the quotient map $mathbb{Z}tomathbb{Z}/pmathbb{Z}$ for prime $p$. But obviously not every field is a homomorphic image of $mathbb{Z}$ (e.g. reals).
– freakish
Nov 20 at 9:11
Sorry but I have not explained well, now I correct my question.
– Jack J.
Nov 20 at 9:18
Sorry but I have not explained well, now I correct my question.
– Jack J.
Nov 20 at 9:18
The kernel cannot be zero, since this would imply that $2$, for exemple, has multiplicative inverse (since its image in F does). Now, by the isomorphism theorem you have $Z/nZ cong F$. Thus, $Z/nZ$ is a field. Now prove that $n$ must be prime.
– math.h
Nov 20 at 10:17
The kernel cannot be zero, since this would imply that $2$, for exemple, has multiplicative inverse (since its image in F does). Now, by the isomorphism theorem you have $Z/nZ cong F$. Thus, $Z/nZ$ is a field. Now prove that $n$ must be prime.
– math.h
Nov 20 at 10:17
@math.h I would like to know if the way I have shown that $n$ can not be zero is correct
– Jack J.
Nov 20 at 10:23
@math.h I would like to know if the way I have shown that $n$ can not be zero is correct
– Jack J.
Nov 20 at 10:23
1
1
Yes, it is. But I think you are "talking too much" (no offense, really) for just a simple thing. The morphism $Zrightarrow F$ cannot be injective (otherwise the morphism would be bijective), since $F$ is a field. You seem to know that $Z$ can't be isomorphic to a field, so you gotta accept this argument. As you said, the kernel should be of the form: $nZ$. As it is not zero (the kernel), then $n$ cannot be zero.
– math.h
Nov 20 at 10:35
Yes, it is. But I think you are "talking too much" (no offense, really) for just a simple thing. The morphism $Zrightarrow F$ cannot be injective (otherwise the morphism would be bijective), since $F$ is a field. You seem to know that $Z$ can't be isomorphic to a field, so you gotta accept this argument. As you said, the kernel should be of the form: $nZ$. As it is not zero (the kernel), then $n$ cannot be zero.
– math.h
Nov 20 at 10:35
|
show 2 more comments
1 Answer
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We can assume $ker(f)neq (0)$, since this would imply that $Bbb Z$ is a field.
Since $Bbb Z$ is a PID, we can assume $ker(f)=(n)$, where $ninBbb Z^+$. You are ofcourse right that $Bbb Z_n$ is a field $iff n$ prime. One way to show this:
Suppose $n=ab$ so that $ker(f)=(ab)$, then $f(ab)=f(a)f(b)=0$ in $F$. Since $F$ is a field, and so an integral domain we need to have $f(a)=0$ or $f(b)=0$, which forces either $a,b$ to be a unit. This means by definition that $n$ is prime.
answered Nov 20 at 9:55
cansomeonehelpmeout
6,4583834
@cansomeonehelpmeoutThanks for yourn answer. So my reasoning to show that $n$ can not be zero is correct?
– Jack J.
Nov 20 at 10:14
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
We can assume $ker(f)neq (0)$, since this would imply that $Bbb Z$ is a field.
Since $Bbb Z$ is a PID, we can assume $ker(f)=(n)$, where $ninBbb Z^+$. You are ofcourse right that $Bbb Z_n$ is a field $iff n$ prime. One way to show this:
Suppose $n=ab$ so that $ker(f)=(ab)$, then $f(ab)=f(a)f(b)=0$ in $F$. Since $F$ is a field, and so an integral domain we need to have $f(a)=0$ or $f(b)=0$, which forces either $a,b$ to be a unit. This means by definition that $n$ is prime.
answered Nov 20 at 9:55
cansomeonehelpmeout
6,4583834
@cansomeonehelpmeoutThanks for yourn answer. So my reasoning to show that $n$ can not be zero is correct?
– Jack J.
Nov 20 at 10:14
add a comment |
up vote
1
down vote
We can assume $ker(f)neq (0)$, since this would imply that $Bbb Z$ is a field.
Since $Bbb Z$ is a PID, we can assume $ker(f)=(n)$, where $ninBbb Z^+$. You are ofcourse right that $Bbb Z_n$ is a field $iff n$ prime. One way to show this:
Suppose $n=ab$ so that $ker(f)=(ab)$, then $f(ab)=f(a)f(b)=0$ in $F$. Since $F$ is a field, and so an integral domain we need to have $f(a)=0$ or $f(b)=0$, which forces either $a,b$ to be a unit. This means by definition that $n$ is prime.
answered Nov 20 at 9:55
cansomeonehelpmeout
6,4583834
@cansomeonehelpmeoutThanks for yourn answer. So my reasoning to show that $n$ can not be zero is correct?
– Jack J.
Nov 20 at 10:14
add a comment |
up vote
1
down vote
up vote
1
down vote
We can assume $ker(f)neq (0)$, since this would imply that $Bbb Z$ is a field.
Since $Bbb Z$ is a PID, we can assume $ker(f)=(n)$, where $ninBbb Z^+$. You are ofcourse right that $Bbb Z_n$ is a field $iff n$ prime. One way to show this:
Suppose $n=ab$ so that $ker(f)=(ab)$, then $f(ab)=f(a)f(b)=0$ in $F$. Since $F$ is a field, and so an integral domain we need to have $f(a)=0$ or $f(b)=0$, which forces either $a,b$ to be a unit. This means by definition that $n$ is prime.
answered Nov 20 at 9:55
cansomeonehelpmeout
6,4583834
We can assume $ker(f)neq (0)$, since this would imply that $Bbb Z$ is a field.
Since $Bbb Z$ is a PID, we can assume $ker(f)=(n)$, where $ninBbb Z^+$. You are ofcourse right that $Bbb Z_n$ is a field $iff n$ prime. One way to show this:
Suppose $n=ab$ so that $ker(f)=(ab)$, then $f(ab)=f(a)f(b)=0$ in $F$. Since $F$ is a field, and so an integral domain we need to have $f(a)=0$ or $f(b)=0$, which forces either $a,b$ to be a unit. This means by definition that $n$ is prime.
answered Nov 20 at 9:55
cansomeonehelpmeout
6,4583834
answered Nov 20 at 9:55
cansomeonehelpmeout
6,4583834
answered Nov 20 at 9:55
cansomeonehelpmeout
6,4583834
answered Nov 20 at 9:55
cansomeonehelpmeout
6,4583834
6,4583834
@cansomeonehelpmeoutThanks for yourn answer. So my reasoning to show that $n$ can not be zero is correct?
– Jack J.
Nov 20 at 10:14
add a comment |
@cansomeonehelpmeoutThanks for yourn answer. So my reasoning to show that $n$ can not be zero is correct?
– Jack J.
Nov 20 at 10:14
@cansomeonehelpmeoutThanks for yourn answer. So my reasoning to show that $n$ can not be zero is correct?
– Jack J.
Nov 20 at 10:14
@cansomeonehelpmeoutThanks for yourn answer. So my reasoning to show that $n$ can not be zero is correct?
– Jack J.
Nov 20 at 10:14
add a comment |
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I don't understand your claim. $mathbb{Z}$ is not isomorphic to a field because it is not a field: $2$ has no multiplicative inverse. And there is an onto homomorphism from $mathbb{Z}$ to a field, take the quotient map $mathbb{Z}tomathbb{Z}/pmathbb{Z}$ for prime $p$. But obviously not every field is a homomorphic image of $mathbb{Z}$ (e.g. reals).
– freakish
Nov 20 at 9:11
Sorry but I have not explained well, now I correct my question.
– Jack J.
Nov 20 at 9:18
The kernel cannot be zero, since this would imply that $2$, for exemple, has multiplicative inverse (since its image in F does). Now, by the isomorphism theorem you have $Z/nZ cong F$. Thus, $Z/nZ$ is a field. Now prove that $n$ must be prime.
– math.h
Nov 20 at 10:17
@math.h I would like to know if the way I have shown that $n$ can not be zero is correct
– Jack J.
Nov 20 at 10:23
1
Yes, it is. But I think you are "talking too much" (no offense, really) for just a simple thing. The morphism $Zrightarrow F$ cannot be injective (otherwise the morphism would be bijective), since $F$ is a field. You seem to know that $Z$ can't be isomorphic to a field, so you gotta accept this argument. As you said, the kernel should be of the form: $nZ$. As it is not zero (the kernel), then $n$ cannot be zero.
– math.h
Nov 20 at 10:35