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Proving $mathbbZ[i]^times$ is finite [duplicate]
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Units of Gaussian integers
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I have no idea how to prove that $mathbbZ[i]^times$ is finite (unit group of $mathbbZ[i]$).
I don't know any theorem and never studied properties of unit groups, so I don't even know where to start here.
This fact fact appeared to me as an example while I was studying modules.
Any help or hint would be appreciated. Thanks in advance.
abstract-algebra group-theory
asked Jul 24 at 6:21
MarinUW
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marked as duplicate by Dietrich Burde abstract-algebra
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Jul 24 at 12:35
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
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This question already has an answer here:
Units of Gaussian integers
3 answers
I have no idea how to prove that $mathbbZ[i]^times$ is finite (unit group of $mathbbZ[i]$).
I don't know any theorem and never studied properties of unit groups, so I don't even know where to start here.
This fact fact appeared to me as an example while I was studying modules.
Any help or hint would be appreciated. Thanks in advance.
abstract-algebra group-theory
asked Jul 24 at 6:21
MarinUW
62
marked as duplicate by Dietrich Burde abstract-algebra
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Jul 24 at 12:35
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
Thanks. And, like I said, I REALLY don't know where to start, I think I never proved that a group is finite in my life.
– MarinUW
Jul 24 at 6:26
1
Consider norm function $N: mathbb Z[i] rightarrow mathbbZ$, $N(a+bi) = a^2 + b^2$... Have you heard of this before?
– cat
Jul 24 at 6:28
add a comment |Â
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up vote
0
down vote
favorite
This question already has an answer here:
Units of Gaussian integers
3 answers
I have no idea how to prove that $mathbbZ[i]^times$ is finite (unit group of $mathbbZ[i]$).
I don't know any theorem and never studied properties of unit groups, so I don't even know where to start here.
This fact fact appeared to me as an example while I was studying modules.
Any help or hint would be appreciated. Thanks in advance.
abstract-algebra group-theory
asked Jul 24 at 6:21
MarinUW
62
This question already has an answer here:
Units of Gaussian integers
3 answers
I have no idea how to prove that $mathbbZ[i]^times$ is finite (unit group of $mathbbZ[i]$).
I don't know any theorem and never studied properties of unit groups, so I don't even know where to start here.
This fact fact appeared to me as an example while I was studying modules.
Any help or hint would be appreciated. Thanks in advance.
This question already has an answer here:
Units of Gaussian integers
3 answers
abstract-algebra group-theory
asked Jul 24 at 6:21
MarinUW
62
asked Jul 24 at 6:21
MarinUW
62
asked Jul 24 at 6:21
MarinUW
62
asked Jul 24 at 6:21
MarinUW
62
62
marked as duplicate by Dietrich Burde abstract-algebra
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Jul 24 at 12:35
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by Dietrich Burde abstract-algebra
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Jul 24 at 12:35
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
Thanks. And, like I said, I REALLY don't know where to start, I think I never proved that a group is finite in my life.
– MarinUW
Jul 24 at 6:26
1
Consider norm function $N: mathbb Z[i] rightarrow mathbbZ$, $N(a+bi) = a^2 + b^2$... Have you heard of this before?
– cat
Jul 24 at 6:28
add a comment |Â
Thanks. And, like I said, I REALLY don't know where to start, I think I never proved that a group is finite in my life.
– MarinUW
Jul 24 at 6:26
1
Consider norm function $N: mathbb Z[i] rightarrow mathbbZ$, $N(a+bi) = a^2 + b^2$... Have you heard of this before?
– cat
Jul 24 at 6:28
Thanks. And, like I said, I REALLY don't know where to start, I think I never proved that a group is finite in my life.
– MarinUW
Jul 24 at 6:26
Thanks. And, like I said, I REALLY don't know where to start, I think I never proved that a group is finite in my life.
– MarinUW
Jul 24 at 6:26
1
1
Consider norm function $N: mathbb Z[i] rightarrow mathbbZ$, $N(a+bi) = a^2 + b^2$... Have you heard of this before?
– cat
Jul 24 at 6:28
Consider norm function $N: mathbb Z[i] rightarrow mathbbZ$, $N(a+bi) = a^2 + b^2$... Have you heard of this before?
– cat
Jul 24 at 6:28
add a comment |Â
1 Answer
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Hint: $frac1a+bi=fraca-bia^2+b^2$. If $a,bin mathbb Z$, when is this still in $mathbb Z[i]$?
answered Jul 24 at 6:32
Aaron
15.1k22552
2
Here is an alternative hint. Define $N(alpha)=alpha overlinealpha$. Then $N(alpha)$ is real, and $N(alphabeta)=N(alpha)N(beta)$. What happens if $alphabeta=1$?
– Aaron
Jul 24 at 7:09
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Hint: $frac1a+bi=fraca-bia^2+b^2$. If $a,bin mathbb Z$, when is this still in $mathbb Z[i]$?
answered Jul 24 at 6:32
Aaron
15.1k22552
2
Here is an alternative hint. Define $N(alpha)=alpha overlinealpha$. Then $N(alpha)$ is real, and $N(alphabeta)=N(alpha)N(beta)$. What happens if $alphabeta=1$?
– Aaron
Jul 24 at 7:09
add a comment |Â
up vote
1
down vote
accepted
Hint: $frac1a+bi=fraca-bia^2+b^2$. If $a,bin mathbb Z$, when is this still in $mathbb Z[i]$?
answered Jul 24 at 6:32
Aaron
15.1k22552
2
Here is an alternative hint. Define $N(alpha)=alpha overlinealpha$. Then $N(alpha)$ is real, and $N(alphabeta)=N(alpha)N(beta)$. What happens if $alphabeta=1$?
– Aaron
Jul 24 at 7:09
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Hint: $frac1a+bi=fraca-bia^2+b^2$. If $a,bin mathbb Z$, when is this still in $mathbb Z[i]$?
answered Jul 24 at 6:32
Aaron
15.1k22552
Hint: $frac1a+bi=fraca-bia^2+b^2$. If $a,bin mathbb Z$, when is this still in $mathbb Z[i]$?
answered Jul 24 at 6:32
Aaron
15.1k22552
answered Jul 24 at 6:32
Aaron
15.1k22552
answered Jul 24 at 6:32
Aaron
15.1k22552
answered Jul 24 at 6:32
Aaron
15.1k22552
15.1k22552
2
Here is an alternative hint. Define $N(alpha)=alpha overlinealpha$. Then $N(alpha)$ is real, and $N(alphabeta)=N(alpha)N(beta)$. What happens if $alphabeta=1$?
– Aaron
Jul 24 at 7:09
add a comment |Â
2
Here is an alternative hint. Define $N(alpha)=alpha overlinealpha$. Then $N(alpha)$ is real, and $N(alphabeta)=N(alpha)N(beta)$. What happens if $alphabeta=1$?
– Aaron
Jul 24 at 7:09
2
2
Here is an alternative hint. Define $N(alpha)=alpha overlinealpha$. Then $N(alpha)$ is real, and $N(alphabeta)=N(alpha)N(beta)$. What happens if $alphabeta=1$?
– Aaron
Jul 24 at 7:09
Here is an alternative hint. Define $N(alpha)=alpha overlinealpha$. Then $N(alpha)$ is real, and $N(alphabeta)=N(alpha)N(beta)$. What happens if $alphabeta=1$?
– Aaron
Jul 24 at 7:09
add a comment |Â
Thanks. And, like I said, I REALLY don't know where to start, I think I never proved that a group is finite in my life.
– MarinUW
Jul 24 at 6:26
1
Consider norm function $N: mathbb Z[i] rightarrow mathbbZ$, $N(a+bi) = a^2 + b^2$... Have you heard of this before?
– cat
Jul 24 at 6:28