Mixing
Find all units in the ring Z[i] = { a+bi : a,b ϵ Z } [duplicate]
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This question already has an answer here:
Units of Gaussian integers
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Find all units in the ring $ Z[i] $= { $a+bi$ : $a,b$ ϵ $Z$ }.
I faced a similar problem to find all the invertible matrices in $Z$. I concluded the solution must be all matrices of det ($pm1$). I don't know whether this solution is correct or accurate but I'm clueless for above problem.
Edit: The chapter was on matrices so I assumed the author was asking for such matrices, but author just said "Find all units" (He may have meant elements, idk).
matrices inverse
asked Jan 21 at 14:59
prakashtprakasht
11
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marked as duplicate by Servaes, José Carlos Santos matrices
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Jan 21 at 15:12
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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$begingroup$
This question already has an answer here:
Units of Gaussian integers
3 answers
Find all units in the ring $ Z[i] $= { $a+bi$ : $a,b$ ϵ $Z$ }.
I faced a similar problem to find all the invertible matrices in $Z$. I concluded the solution must be all matrices of det ($pm1$). I don't know whether this solution is correct or accurate but I'm clueless for above problem.
Edit: The chapter was on matrices so I assumed the author was asking for such matrices, but author just said "Find all units" (He may have meant elements, idk).
matrices inverse
asked Jan 21 at 14:59
prakashtprakasht
11
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marked as duplicate by Servaes, José Carlos Santos matrices
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Jan 21 at 15:12
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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Welcome to stackexchange. Where are there matrices and determinants in these objects? Please edit the question to clarify, and show us what you did for $mathbb{Z}$.
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– Ethan Bolker
Jan 21 at 15:05
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Thanks a lot, the above mentioned question "Units of Gaussian integers" solved my doubts:p
$endgroup$
– prakasht
Jan 21 at 15:24
add a comment |
$begingroup$
This question already has an answer here:
Units of Gaussian integers
3 answers
Find all units in the ring $ Z[i] $= { $a+bi$ : $a,b$ ϵ $Z$ }.
I faced a similar problem to find all the invertible matrices in $Z$. I concluded the solution must be all matrices of det ($pm1$). I don't know whether this solution is correct or accurate but I'm clueless for above problem.
Edit: The chapter was on matrices so I assumed the author was asking for such matrices, but author just said "Find all units" (He may have meant elements, idk).
matrices inverse
asked Jan 21 at 14:59
prakashtprakasht
11
$endgroup$
This question already has an answer here:
Units of Gaussian integers
3 answers
Find all units in the ring $ Z[i] $= { $a+bi$ : $a,b$ ϵ $Z$ }.
I faced a similar problem to find all the invertible matrices in $Z$. I concluded the solution must be all matrices of det ($pm1$). I don't know whether this solution is correct or accurate but I'm clueless for above problem.
Edit: The chapter was on matrices so I assumed the author was asking for such matrices, but author just said "Find all units" (He may have meant elements, idk).
This question already has an answer here:
Units of Gaussian integers
3 answers
matrices inverse
matrices inverse
asked Jan 21 at 14:59
prakashtprakasht
11
asked Jan 21 at 14:59
prakashtprakasht
11
edited Jan 21 at 15:18
prakasht
asked Jan 21 at 14:59
prakashtprakasht
11
asked Jan 21 at 14:59
prakashtprakasht
11
asked Jan 21 at 14:59
prakashtprakasht
11
11
marked as duplicate by Servaes, José Carlos Santos matrices
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Jan 21 at 15:12
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Jan 21 at 15:12
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.
2
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Welcome to stackexchange. Where are there matrices and determinants in these objects? Please edit the question to clarify, and show us what you did for $mathbb{Z}$.
$endgroup$
– Ethan Bolker
Jan 21 at 15:05
$begingroup$
Thanks a lot, the above mentioned question "Units of Gaussian integers" solved my doubts:p
$endgroup$
– prakasht
Jan 21 at 15:24
add a comment |
2
$begingroup$
Welcome to stackexchange. Where are there matrices and determinants in these objects? Please edit the question to clarify, and show us what you did for $mathbb{Z}$.
$endgroup$
– Ethan Bolker
Jan 21 at 15:05
$begingroup$
Thanks a lot, the above mentioned question "Units of Gaussian integers" solved my doubts:p
$endgroup$
– prakasht
Jan 21 at 15:24
2
2
$begingroup$
Welcome to stackexchange. Where are there matrices and determinants in these objects? Please edit the question to clarify, and show us what you did for $mathbb{Z}$.
$endgroup$
– Ethan Bolker
Jan 21 at 15:05
$begingroup$
Welcome to stackexchange. Where are there matrices and determinants in these objects? Please edit the question to clarify, and show us what you did for $mathbb{Z}$.
$endgroup$
– Ethan Bolker
Jan 21 at 15:05
$begingroup$
Thanks a lot, the above mentioned question "Units of Gaussian integers" solved my doubts:p
$endgroup$
– prakasht
Jan 21 at 15:24
$begingroup$
Thanks a lot, the above mentioned question "Units of Gaussian integers" solved my doubts:p
$endgroup$
– prakasht
Jan 21 at 15:24
add a comment |
1 Answer
1
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You are right. With $$1=begin{pmatrix}1&0\0&1end{pmatrix}, i=begin{pmatrix}0&-1\1&0end{pmatrix}$$ you can write any $a+bimid a,binBbb Z$ as $$begin{pmatrix}a&-b\b&aend{pmatrix}$$
That is $Bbb Z[i]congleft{begin{pmatrix}a&-b\b&aend{pmatrix}mid a,binBbb Zright}$. Can you take it from here?
answered Jan 21 at 15:07
cansomeonehelpmeoutcansomeonehelpmeout
7,0973935
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add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
You are right. With $$1=begin{pmatrix}1&0\0&1end{pmatrix}, i=begin{pmatrix}0&-1\1&0end{pmatrix}$$ you can write any $a+bimid a,binBbb Z$ as $$begin{pmatrix}a&-b\b&aend{pmatrix}$$
That is $Bbb Z[i]congleft{begin{pmatrix}a&-b\b&aend{pmatrix}mid a,binBbb Zright}$. Can you take it from here?
answered Jan 21 at 15:07
cansomeonehelpmeoutcansomeonehelpmeout
7,0973935
$endgroup$
add a comment |
$begingroup$
You are right. With $$1=begin{pmatrix}1&0\0&1end{pmatrix}, i=begin{pmatrix}0&-1\1&0end{pmatrix}$$ you can write any $a+bimid a,binBbb Z$ as $$begin{pmatrix}a&-b\b&aend{pmatrix}$$
That is $Bbb Z[i]congleft{begin{pmatrix}a&-b\b&aend{pmatrix}mid a,binBbb Zright}$. Can you take it from here?
answered Jan 21 at 15:07
cansomeonehelpmeoutcansomeonehelpmeout
7,0973935
$endgroup$
add a comment |
$begingroup$
You are right. With $$1=begin{pmatrix}1&0\0&1end{pmatrix}, i=begin{pmatrix}0&-1\1&0end{pmatrix}$$ you can write any $a+bimid a,binBbb Z$ as $$begin{pmatrix}a&-b\b&aend{pmatrix}$$
That is $Bbb Z[i]congleft{begin{pmatrix}a&-b\b&aend{pmatrix}mid a,binBbb Zright}$. Can you take it from here?
answered Jan 21 at 15:07
cansomeonehelpmeoutcansomeonehelpmeout
7,0973935
$endgroup$
You are right. With $$1=begin{pmatrix}1&0\0&1end{pmatrix}, i=begin{pmatrix}0&-1\1&0end{pmatrix}$$ you can write any $a+bimid a,binBbb Z$ as $$begin{pmatrix}a&-b\b&aend{pmatrix}$$
That is $Bbb Z[i]congleft{begin{pmatrix}a&-b\b&aend{pmatrix}mid a,binBbb Zright}$. Can you take it from here?
answered Jan 21 at 15:07
cansomeonehelpmeoutcansomeonehelpmeout
7,0973935
answered Jan 21 at 15:07
cansomeonehelpmeoutcansomeonehelpmeout
7,0973935
answered Jan 21 at 15:07
cansomeonehelpmeoutcansomeonehelpmeout
7,0973935
answered Jan 21 at 15:07
cansomeonehelpmeoutcansomeonehelpmeout
7,0973935
7,0973935
add a comment |
add a comment |
2
$begingroup$
Welcome to stackexchange. Where are there matrices and determinants in these objects? Please edit the question to clarify, and show us what you did for $mathbb{Z}$.
$endgroup$
– Ethan Bolker
Jan 21 at 15:05
$begingroup$
Thanks a lot, the above mentioned question "Units of Gaussian integers" solved my doubts:p
$endgroup$
– prakasht
Jan 21 at 15:24