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Transcendence degree of quotient field of polynomial ring over a field

Can a subquotient field of a field have a higher transcendence degree?Transcendence degree of a field extensionSources about transcendence degreeQuotient of polynomial ring in two variables is a PIDTranscendence Degree of a field extension over $mathbb C$Isomorphic quotient rings of polynomial rings over fieldTwo field extensions of transcendence degree 1Number of roots of a polynomial over a commutative ringQuestion regarding transcendence degreedetermining if quotient ring of polynomials over a finite field is a field or not

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Let $k$ be a field and $A$ the polynomial ring in $n$ variables over $k$. How do you determine the transcendence degree of the quotient field $K(A) $ of $A$ over $k$ ? Thanks.

abstract-algebra

share|cite|improve this question

asked Mar 12 at 10:53

user249018user249018

435137

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  Usually by finding a transcendence basis, which should not be very hard in this case.
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  – asdq
  Mar 12 at 11:08
  

  
  
  
  


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  The transcendence degree of this extension is by definition the cardinality of a transcendence basis of $K(A)/k$. So your task is to find such a basis. Note that this is not the same as a vector space basis.
  $endgroup$
  – asdq
  Mar 12 at 12:45
  

  
  
  
  


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  I don't understand what you mean by ideal in this context. $K(A)$ is generated as a field over $k$ by the indeterminates $x_1,dots,x_n$ of the polynomial ring $A=k[x_1,dots,x_n]$, and these are algebraically independent by definition, hence form a transcendence basis of $K(A)$ over $k$.
  $endgroup$
  – asdq
  Mar 12 at 15:24
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
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  Note that in order to show that these elements indeed generate $K(A)$ as a field, it suffices to show that they generate the polynomial ring $A$ as an algebra over $k$ (this follows from the universal property of the field of fractions).
  $endgroup$
  – asdq
  Mar 12 at 15:27
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Transcendence degree is always understood with respect to a field extension, you should have a look at the definition again (see my previous comment).
  $endgroup$
  – asdq
  Mar 12 at 17:04
  

  
  
  
  

 | 
show 7 more comments

0

$begingroup$

Let $k$ be a field and $A$ the polynomial ring in $n$ variables over $k$. How do you determine the transcendence degree of the quotient field $K(A) $ of $A$ over $k$ ? Thanks.

abstract-algebra

share|cite|improve this question

asked Mar 12 at 10:53

user249018user249018

435137

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Usually by finding a transcendence basis, which should not be very hard in this case.
  $endgroup$
  – asdq
  Mar 12 at 11:08
  

  
  
  
  


• 
  
  
  

  
  3
  

  
  
  
  
  
  

  
  $begingroup$
  The transcendence degree of this extension is by definition the cardinality of a transcendence basis of $K(A)/k$. So your task is to find such a basis. Note that this is not the same as a vector space basis.
  $endgroup$
  – asdq
  Mar 12 at 12:45
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I don't understand what you mean by ideal in this context. $K(A)$ is generated as a field over $k$ by the indeterminates $x_1,dots,x_n$ of the polynomial ring $A=k[x_1,dots,x_n]$, and these are algebraically independent by definition, hence form a transcendence basis of $K(A)$ over $k$.
  $endgroup$
  – asdq
  Mar 12 at 15:24
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Note that in order to show that these elements indeed generate $K(A)$ as a field, it suffices to show that they generate the polynomial ring $A$ as an algebra over $k$ (this follows from the universal property of the field of fractions).
  $endgroup$
  – asdq
  Mar 12 at 15:27
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  Transcendence degree is always understood with respect to a field extension, you should have a look at the definition again (see my previous comment).
  $endgroup$
  – asdq
  Mar 12 at 17:04
  

  
  
  
  

 | 
show 7 more comments

$begingroup$

Let $k$ be a field and $A$ the polynomial ring in $n$ variables over $k$. How do you determine the transcendence degree of the quotient field $K(A) $ of $A$ over $k$ ? Thanks.

abstract-algebra

share|cite|improve this question

asked Mar 12 at 10:53

user249018user249018

435137

$endgroup$

share|cite|improve this question

asked Mar 12 at 10:53

user249018user249018

435137

share|cite|improve this question

asked Mar 12 at 10:53

user249018user249018

435137

asked Mar 12 at 10:53

user249018user249018

435137

asked Mar 12 at 10:53

user249018user249018

435137