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Equivalent definitions of separable extension of a field

0

2

$begingroup$

Armand Borel in his textbook "Linear Algebraic Groups", pp. 4, states that:

$F$ is said to be separable over $boldsymbolk$ if it satisfies the following equivalent conditions ($p$ denotes the characteristic exponent of $k$):

(1) $F^p$ and $k$ are linearly disjoint over $k^p$.

(2) $(k^1/p)otimes_k F$ is reduced.

(3) $k'otimes_k F$ is reduced for all field estensions $k'$ of $k$.

My question is: how can I prove this equivalence?
I also gave a look at separable extension, but I really don't know how to proceed maybe because I don't have understood what I need to prove.

My background: I have never attended a course in algebraic geometry and I have a basic knowledge of field theory. Moreover, any suggestion about preliminary texts I need to study is appreciated.

abstract-algebra algebraic-geometry tensor-products algebraic-groups separable-extension

share|cite|improve this question

edited Mar 25 at 11:18

LBJFS

asked Mar 24 at 21:06

LBJFSLBJFS

372112

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Your post does not include a question. What are you trying to do? Are you attempting to prove the equivalence of the given conditions?
  $endgroup$
  – KReiser
  Mar 25 at 1:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @KReiser Noi are right. I was so tired that I forgot to include the question. Now I edit
  $endgroup$
  – LBJFS
  Mar 25 at 6:50
  
  

  
  
  
  

add a comment | 

0

2

$begingroup$

Armand Borel in his textbook "Linear Algebraic Groups", pp. 4, states that:

$F$ is said to be separable over $boldsymbolk$ if it satisfies the following equivalent conditions ($p$ denotes the characteristic exponent of $k$):

(1) $F^p$ and $k$ are linearly disjoint over $k^p$.

(2) $(k^1/p)otimes_k F$ is reduced.

(3) $k'otimes_k F$ is reduced for all field estensions $k'$ of $k$.

My question is: how can I prove this equivalence?
I also gave a look at separable extension, but I really don't know how to proceed maybe because I don't have understood what I need to prove.

My background: I have never attended a course in algebraic geometry and I have a basic knowledge of field theory. Moreover, any suggestion about preliminary texts I need to study is appreciated.

abstract-algebra algebraic-geometry tensor-products algebraic-groups separable-extension

share|cite|improve this question

edited Mar 25 at 11:18

LBJFS

asked Mar 24 at 21:06

LBJFSLBJFS

372112

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Your post does not include a question. What are you trying to do? Are you attempting to prove the equivalence of the given conditions?
  $endgroup$
  – KReiser
  Mar 25 at 1:32
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @KReiser Noi are right. I was so tired that I forgot to include the question. Now I edit
  $endgroup$
  – LBJFS
  Mar 25 at 6:50
  
  

  
  
  
  

add a comment | 

$begingroup$

Armand Borel in his textbook "Linear Algebraic Groups", pp. 4, states that:

$F$ is said to be separable over $boldsymbolk$ if it satisfies the following equivalent conditions ($p$ denotes the characteristic exponent of $k$):

(1) $F^p$ and $k$ are linearly disjoint over $k^p$.

(2) $(k^1/p)otimes_k F$ is reduced.

(3) $k'otimes_k F$ is reduced for all field estensions $k'$ of $k$.

My question is: how can I prove this equivalence?
I also gave a look at separable extension, but I really don't know how to proceed maybe because I don't have understood what I need to prove.

My background: I have never attended a course in algebraic geometry and I have a basic knowledge of field theory. Moreover, any suggestion about preliminary texts I need to study is appreciated.

abstract-algebra algebraic-geometry tensor-products algebraic-groups separable-extension

share|cite|improve this question

edited Mar 25 at 11:18

LBJFS

asked Mar 24 at 21:06

LBJFSLBJFS

372112

$endgroup$

share|cite|improve this question

edited Mar 25 at 11:18

LBJFS

asked Mar 24 at 21:06

LBJFSLBJFS

372112

share|cite|improve this question

edited Mar 25 at 11:18

LBJFS

asked Mar 24 at 21:06

LBJFSLBJFS

372112

asked Mar 24 at 21:06

LBJFSLBJFS

372112

asked Mar 24 at 21:06

LBJFSLBJFS

372112