Is $frac{ mathbb{C}[x,y] }{langle x^m+ y^n -1rangle} cong frac{ mathbb{C}[y]}{langle y^n-1rangle }$?...
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Today I saw an isomorphism in my friends notebook. It was written as $$frac{ mathbb{C}[x,y] }{langle x^m+ y^n -1rangle} cong frac{ mathbb{C}[y]}{langle y^n-1rangle }.$$
I want to confirm that it is true/false. I'm very confused as to how to do this.
Any hints about a solution ?
abstract-algebra
edited Jan 22 at 22:18
Leucippus
19.8k102871
asked Jan 15 at 10:28
jasminejasmine
1,988420
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closed as off-topic by Lord_Farin, Did, onurcanbektas, Lee David Chung Lin, Gibbs Jan 22 at 10:36
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Lord_Farin, Did, onurcanbektas, Lee David Chung Lin, Gibbs
If this question can be reworded to fit the rules in the help center, please edit the question.
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$begingroup$
Today I saw an isomorphism in my friends notebook. It was written as $$frac{ mathbb{C}[x,y] }{langle x^m+ y^n -1rangle} cong frac{ mathbb{C}[y]}{langle y^n-1rangle }.$$
I want to confirm that it is true/false. I'm very confused as to how to do this.
Any hints about a solution ?
abstract-algebra
edited Jan 22 at 22:18
Leucippus
19.8k102871
asked Jan 15 at 10:28
jasminejasmine
1,988420
$endgroup$
closed as off-topic by Lord_Farin, Did, onurcanbektas, Lee David Chung Lin, Gibbs Jan 22 at 10:36
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Lord_Farin, Did, onurcanbektas, Lee David Chung Lin, Gibbs
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
Today I saw an isomorphism in my friends notebook. It was written as $$frac{ mathbb{C}[x,y] }{langle x^m+ y^n -1rangle} cong frac{ mathbb{C}[y]}{langle y^n-1rangle }.$$
I want to confirm that it is true/false. I'm very confused as to how to do this.
Any hints about a solution ?
abstract-algebra
edited Jan 22 at 22:18
Leucippus
19.8k102871
asked Jan 15 at 10:28
jasminejasmine
1,988420
$endgroup$
Today I saw an isomorphism in my friends notebook. It was written as $$frac{ mathbb{C}[x,y] }{langle x^m+ y^n -1rangle} cong frac{ mathbb{C}[y]}{langle y^n-1rangle }.$$
I want to confirm that it is true/false. I'm very confused as to how to do this.
Any hints about a solution ?
abstract-algebra
abstract-algebra
edited Jan 22 at 22:18
Leucippus
19.8k102871
asked Jan 15 at 10:28
jasminejasmine
1,988420
edited Jan 22 at 22:18
Leucippus
19.8k102871
asked Jan 15 at 10:28
jasminejasmine
1,988420
edited Jan 22 at 22:18
Leucippus
19.8k102871
edited Jan 22 at 22:18
Leucippus
19.8k102871
edited Jan 22 at 22:18
Leucippus
19.8k102871
19.8k102871
asked Jan 15 at 10:28
jasminejasmine
1,988420
asked Jan 15 at 10:28
jasminejasmine
1,988420
asked Jan 15 at 10:28
jasminejasmine
1,988420
1,988420
closed as off-topic by Lord_Farin, Did, onurcanbektas, Lee David Chung Lin, Gibbs Jan 22 at 10:36
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Lord_Farin, Did, onurcanbektas, Lee David Chung Lin, Gibbs
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Lord_Farin, Did, onurcanbektas, Lee David Chung Lin, Gibbs Jan 22 at 10:36
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Lord_Farin, Did, onurcanbektas, Lee David Chung Lin, Gibbs
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
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1 Answer
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Hint: The residue class ring ${Bbb C}[y]/langle y^n-1rangle$ has the ${Bbb C}$-basis ${1+I,y+I,ldots,y^{n-1}+I}$, where $I=langle y^n-1rangle$.
The situation is different for the residue class ring ${Bbb C}[x,y]/langle x^m+y^n-1rangle$. Put $J=langle x^m+y^n-1rangle$. Here you have $x^m+J = (-y^n+1)+J$. So each power of $x^k$ with $kgeq m$ can be replaced by an expression in $y$. Up to the trivial case, the ${Bbb C}$-basis of the residue class ring is definitely larger. Construct the basis...
answered Jan 15 at 12:29
WuestenfuxWuestenfux
5,5911513
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint: The residue class ring ${Bbb C}[y]/langle y^n-1rangle$ has the ${Bbb C}$-basis ${1+I,y+I,ldots,y^{n-1}+I}$, where $I=langle y^n-1rangle$.
The situation is different for the residue class ring ${Bbb C}[x,y]/langle x^m+y^n-1rangle$. Put $J=langle x^m+y^n-1rangle$. Here you have $x^m+J = (-y^n+1)+J$. So each power of $x^k$ with $kgeq m$ can be replaced by an expression in $y$. Up to the trivial case, the ${Bbb C}$-basis of the residue class ring is definitely larger. Construct the basis...
answered Jan 15 at 12:29
WuestenfuxWuestenfux
5,5911513
$endgroup$
add a comment |
$begingroup$
Hint: The residue class ring ${Bbb C}[y]/langle y^n-1rangle$ has the ${Bbb C}$-basis ${1+I,y+I,ldots,y^{n-1}+I}$, where $I=langle y^n-1rangle$.
The situation is different for the residue class ring ${Bbb C}[x,y]/langle x^m+y^n-1rangle$. Put $J=langle x^m+y^n-1rangle$. Here you have $x^m+J = (-y^n+1)+J$. So each power of $x^k$ with $kgeq m$ can be replaced by an expression in $y$. Up to the trivial case, the ${Bbb C}$-basis of the residue class ring is definitely larger. Construct the basis...
answered Jan 15 at 12:29
WuestenfuxWuestenfux
5,5911513
$endgroup$
add a comment |
$begingroup$
Hint: The residue class ring ${Bbb C}[y]/langle y^n-1rangle$ has the ${Bbb C}$-basis ${1+I,y+I,ldots,y^{n-1}+I}$, where $I=langle y^n-1rangle$.
The situation is different for the residue class ring ${Bbb C}[x,y]/langle x^m+y^n-1rangle$. Put $J=langle x^m+y^n-1rangle$. Here you have $x^m+J = (-y^n+1)+J$. So each power of $x^k$ with $kgeq m$ can be replaced by an expression in $y$. Up to the trivial case, the ${Bbb C}$-basis of the residue class ring is definitely larger. Construct the basis...
answered Jan 15 at 12:29
WuestenfuxWuestenfux
5,5911513
$endgroup$
Hint: The residue class ring ${Bbb C}[y]/langle y^n-1rangle$ has the ${Bbb C}$-basis ${1+I,y+I,ldots,y^{n-1}+I}$, where $I=langle y^n-1rangle$.
The situation is different for the residue class ring ${Bbb C}[x,y]/langle x^m+y^n-1rangle$. Put $J=langle x^m+y^n-1rangle$. Here you have $x^m+J = (-y^n+1)+J$. So each power of $x^k$ with $kgeq m$ can be replaced by an expression in $y$. Up to the trivial case, the ${Bbb C}$-basis of the residue class ring is definitely larger. Construct the basis...
answered Jan 15 at 12:29
WuestenfuxWuestenfux
5,5911513
answered Jan 15 at 12:29
WuestenfuxWuestenfux
5,5911513
answered Jan 15 at 12:29
WuestenfuxWuestenfux
5,5911513
answered Jan 15 at 12:29
WuestenfuxWuestenfux
5,5911513
5,5911513
add a comment |
add a comment |
