Mixing
Number field tensor $mathbb{Q}$ isomorphisms. [closed]
1
$begingroup$
Let $mathbb{K}$ be a number field, i.e. a finite extension of $mathbb{Q}$. Consider the tensor product $mathbb{K} otimes mathbb{Q}$.
I am not familiar with tensor products, so I have these questions:
1) $mathbb{K} otimes_{mathbb{K}} mathbb{Q}$ is isomorphic to what?
2) $mathbb{K} otimes_{mathbb{R}} mathbb{Q}$ is isomorphic to what?
3) $mathbb{K} otimes_{mathbb{Q}} mathbb{Q}$ is isomorphic to what?
abstract-algebra algebraic-number-theory tensor-products
edited Jan 19 at 20:31
user26857
39.3k124183
asked Jan 19 at 11:21
C.S.C.S.
235
$endgroup$
closed as off-topic by user26857, Lee David Chung Lin, Cesareo, A. Pongrácz, José Carlos Santos Jan 20 at 16:46
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." – user26857, Lee David Chung Lin, Cesareo, A. Pongrácz, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
1
$begingroup$
Let $mathbb{K}$ be a number field, i.e. a finite extension of $mathbb{Q}$. Consider the tensor product $mathbb{K} otimes mathbb{Q}$.
I am not familiar with tensor products, so I have these questions:
1) $mathbb{K} otimes_{mathbb{K}} mathbb{Q}$ is isomorphic to what?
2) $mathbb{K} otimes_{mathbb{R}} mathbb{Q}$ is isomorphic to what?
3) $mathbb{K} otimes_{mathbb{Q}} mathbb{Q}$ is isomorphic to what?
abstract-algebra algebraic-number-theory tensor-products
edited Jan 19 at 20:31
user26857
39.3k124183
asked Jan 19 at 11:21
C.S.C.S.
235
$endgroup$
closed as off-topic by user26857, Lee David Chung Lin, Cesareo, A. Pongrácz, José Carlos Santos Jan 20 at 16:46
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." – user26857, Lee David Chung Lin, Cesareo, A. Pongrácz, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.
5
$begingroup$
I don't see how the first 2 can be defined
$endgroup$
– sss89
Jan 19 at 11:44
4
$begingroup$
As for the last one, it is isomorphic to $ mathbb{K}$. This is true for any module over any (commutative) ring. It appears in every standard text on commutative algebra, for example, Atiya McDonald, prop. 2.14(iv).
$endgroup$
– sss89
Jan 19 at 11:48
$begingroup$
@sss89 Thank you. Can you please tell me $mathbb{K}otimes_{mathbb{Q}} mathbb{R}$ is isomorphic to what? Does it contains $mathbb{K}$?
$endgroup$
– C.S.
Jan 20 at 12:26
$begingroup$
As a vector space I think it should be isomorphic to $mathbb{R} $ for dimension reason. I guess you are more interested in what is it as a $mathbb{Q} $-algebra. In this case suppose that $mathbb{K} =mathbb{Q}[alpha_1,dots,alpha_r]$. Then $mathbb{K}otimes mathbb{R} =mathbb{R}[alpha_1,dots,alpha_r]$.
$endgroup$
– sss89
Jan 20 at 12:44
add a comment |
1
1
1
$begingroup$
Let $mathbb{K}$ be a number field, i.e. a finite extension of $mathbb{Q}$. Consider the tensor product $mathbb{K} otimes mathbb{Q}$.
I am not familiar with tensor products, so I have these questions:
1) $mathbb{K} otimes_{mathbb{K}} mathbb{Q}$ is isomorphic to what?
2) $mathbb{K} otimes_{mathbb{R}} mathbb{Q}$ is isomorphic to what?
3) $mathbb{K} otimes_{mathbb{Q}} mathbb{Q}$ is isomorphic to what?
abstract-algebra algebraic-number-theory tensor-products
edited Jan 19 at 20:31
user26857
39.3k124183
asked Jan 19 at 11:21
C.S.C.S.
235
$endgroup$
Let $mathbb{K}$ be a number field, i.e. a finite extension of $mathbb{Q}$. Consider the tensor product $mathbb{K} otimes mathbb{Q}$.
I am not familiar with tensor products, so I have these questions:
1) $mathbb{K} otimes_{mathbb{K}} mathbb{Q}$ is isomorphic to what?
2) $mathbb{K} otimes_{mathbb{R}} mathbb{Q}$ is isomorphic to what?
3) $mathbb{K} otimes_{mathbb{Q}} mathbb{Q}$ is isomorphic to what?
abstract-algebra algebraic-number-theory tensor-products
abstract-algebra algebraic-number-theory tensor-products
edited Jan 19 at 20:31
user26857
39.3k124183
asked Jan 19 at 11:21
C.S.C.S.
235
edited Jan 19 at 20:31
user26857
39.3k124183
asked Jan 19 at 11:21
C.S.C.S.
235
edited Jan 19 at 20:31
user26857
39.3k124183
edited Jan 19 at 20:31
user26857
39.3k124183
edited Jan 19 at 20:31
user26857
39.3k124183
39.3k124183
asked Jan 19 at 11:21
C.S.C.S.
235
asked Jan 19 at 11:21
C.S.C.S.
235
asked Jan 19 at 11:21
C.S.C.S.
235
235
closed as off-topic by user26857, Lee David Chung Lin, Cesareo, A. Pongrácz, José Carlos Santos Jan 20 at 16:46
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." – user26857, Lee David Chung Lin, Cesareo, A. Pongrácz, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by user26857, Lee David Chung Lin, Cesareo, A. Pongrácz, José Carlos Santos Jan 20 at 16:46
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." – user26857, Lee David Chung Lin, Cesareo, A. Pongrácz, José Carlos Santos
If this question can be reworded to fit the rules in the help center, please edit the question.
5
$begingroup$
I don't see how the first 2 can be defined
$endgroup$
– sss89
Jan 19 at 11:44
4
$begingroup$
As for the last one, it is isomorphic to $ mathbb{K}$. This is true for any module over any (commutative) ring. It appears in every standard text on commutative algebra, for example, Atiya McDonald, prop. 2.14(iv).
$endgroup$
– sss89
Jan 19 at 11:48
$begingroup$
@sss89 Thank you. Can you please tell me $mathbb{K}otimes_{mathbb{Q}} mathbb{R}$ is isomorphic to what? Does it contains $mathbb{K}$?
$endgroup$
– C.S.
Jan 20 at 12:26
$begingroup$
As a vector space I think it should be isomorphic to $mathbb{R} $ for dimension reason. I guess you are more interested in what is it as a $mathbb{Q} $-algebra. In this case suppose that $mathbb{K} =mathbb{Q}[alpha_1,dots,alpha_r]$. Then $mathbb{K}otimes mathbb{R} =mathbb{R}[alpha_1,dots,alpha_r]$.
$endgroup$
– sss89
Jan 20 at 12:44
add a comment |
5
$begingroup$
I don't see how the first 2 can be defined
$endgroup$
– sss89
Jan 19 at 11:44
4
$begingroup$
As for the last one, it is isomorphic to $ mathbb{K}$. This is true for any module over any (commutative) ring. It appears in every standard text on commutative algebra, for example, Atiya McDonald, prop. 2.14(iv).
$endgroup$
– sss89
Jan 19 at 11:48
$begingroup$
@sss89 Thank you. Can you please tell me $mathbb{K}otimes_{mathbb{Q}} mathbb{R}$ is isomorphic to what? Does it contains $mathbb{K}$?
$endgroup$
– C.S.
Jan 20 at 12:26
$begingroup$
As a vector space I think it should be isomorphic to $mathbb{R} $ for dimension reason. I guess you are more interested in what is it as a $mathbb{Q} $-algebra. In this case suppose that $mathbb{K} =mathbb{Q}[alpha_1,dots,alpha_r]$. Then $mathbb{K}otimes mathbb{R} =mathbb{R}[alpha_1,dots,alpha_r]$.
$endgroup$
– sss89
Jan 20 at 12:44
5
5
$begingroup$
I don't see how the first 2 can be defined
$endgroup$
– sss89
Jan 19 at 11:44
$begingroup$
I don't see how the first 2 can be defined
$endgroup$
– sss89
Jan 19 at 11:44
4
4
$begingroup$
As for the last one, it is isomorphic to $ mathbb{K}$. This is true for any module over any (commutative) ring. It appears in every standard text on commutative algebra, for example, Atiya McDonald, prop. 2.14(iv).
$endgroup$
– sss89
Jan 19 at 11:48
$begingroup$
As for the last one, it is isomorphic to $ mathbb{K}$. This is true for any module over any (commutative) ring. It appears in every standard text on commutative algebra, for example, Atiya McDonald, prop. 2.14(iv).
$endgroup$
– sss89
Jan 19 at 11:48
$begingroup$
@sss89 Thank you. Can you please tell me $mathbb{K}otimes_{mathbb{Q}} mathbb{R}$ is isomorphic to what? Does it contains $mathbb{K}$?
$endgroup$
– C.S.
Jan 20 at 12:26
$begingroup$
@sss89 Thank you. Can you please tell me $mathbb{K}otimes_{mathbb{Q}} mathbb{R}$ is isomorphic to what? Does it contains $mathbb{K}$?
$endgroup$
– C.S.
Jan 20 at 12:26
$begingroup$
As a vector space I think it should be isomorphic to $mathbb{R} $ for dimension reason. I guess you are more interested in what is it as a $mathbb{Q} $-algebra. In this case suppose that $mathbb{K} =mathbb{Q}[alpha_1,dots,alpha_r]$. Then $mathbb{K}otimes mathbb{R} =mathbb{R}[alpha_1,dots,alpha_r]$.
$endgroup$
– sss89
Jan 20 at 12:44
$begingroup$
As a vector space I think it should be isomorphic to $mathbb{R} $ for dimension reason. I guess you are more interested in what is it as a $mathbb{Q} $-algebra. In this case suppose that $mathbb{K} =mathbb{Q}[alpha_1,dots,alpha_r]$. Then $mathbb{K}otimes mathbb{R} =mathbb{R}[alpha_1,dots,alpha_r]$.
$endgroup$
– sss89
Jan 20 at 12:44
add a comment |
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5
$begingroup$
I don't see how the first 2 can be defined
$endgroup$
– sss89
Jan 19 at 11:44
4
$begingroup$
As for the last one, it is isomorphic to $ mathbb{K}$. This is true for any module over any (commutative) ring. It appears in every standard text on commutative algebra, for example, Atiya McDonald, prop. 2.14(iv).
$endgroup$
– sss89
Jan 19 at 11:48
$begingroup$
@sss89 Thank you. Can you please tell me $mathbb{K}otimes_{mathbb{Q}} mathbb{R}$ is isomorphic to what? Does it contains $mathbb{K}$?
$endgroup$
– C.S.
Jan 20 at 12:26
$begingroup$
As a vector space I think it should be isomorphic to $mathbb{R} $ for dimension reason. I guess you are more interested in what is it as a $mathbb{Q} $-algebra. In this case suppose that $mathbb{K} =mathbb{Q}[alpha_1,dots,alpha_r]$. Then $mathbb{K}otimes mathbb{R} =mathbb{R}[alpha_1,dots,alpha_r]$.
$endgroup$
– sss89
Jan 20 at 12:44