[<< wikibooks] Commutative Algebra/Noether's normalisation lemma
== Computational preparation ==
Lemma 23.1:
Let 
  
    
      
        R
      
    
    {\displaystyle R}
   be a ring, and let 
  
    
      
        f
        ∈
        R
        [
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        ]
      
    
    {\displaystyle f\in R[x_{1},\ldots ,x_{n}]}
   be a polynomial. Let 
  
    
      
        N
        ∈
        
          N
        
      
    
    {\displaystyle N\in \mathbb {N} }
   be a number that is strictly larger than the degree of any monomial of 
  
    
      
        f
      
    
    {\displaystyle f}
   (where the degree of an arbitrary monomial 
  
    
      
        
          x
          
            1
          
          
            
              k
              
                1
              
            
          
        
        
          x
          
            2
          
          
            
              k
              
                2
              
            
          
        
        ⋯
        
          x
          
            n
          
          
            
              k
              
                n
              
            
          
        
      
    
    {\displaystyle x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{n}^{k_{n}}}
   of 
  
    
      
        f
      
    
    {\displaystyle f}
   is defined to be 
  
    
      
        
          k
          
            1
          
        
        +
        
          k
          
            2
          
        
        +
        ⋯
        +
        
          k
          
            n
          
        
      
    
    {\displaystyle k_{1}+k_{2}+\cdots +k_{n}}
  ). Then the largest monomial (with respect to degree) of the polynomial

  
    
      
        g
        (
        
          x
          
            1
          
        
        ,
        …
        ,
        
          x
          
            n
          
        
        )
        :=
        f
        (
        
          x
          
            1
          
        
        +
        
          x
          
            n
          
          
            
              N
              
                n
                −
                1
              
            
          
        
        ,
        
          x
          
            2
          
        
        +
        
          x
          
            n
          
          
            
              N
              
                n
                −
                2
              
            
          
        
        ,
        …
        ,
        
          x
          
            n
            −
            2
          
        
        +
        
          x
          
            n
          
          
            
              N
              
                2
              
            
          
        
        ,
        
          x
          
            n
            −
            1
          
        
        +
        
          x
          
            n
          
          
            N
          
        
        ,
        
          x
          
            n
          
        
        )
      
    
    {\displaystyle g(x_{1},\ldots ,x_{n}):=f(x_{1}+x_{n}^{N^{n-1}},x_{2}+x_{n}^{N^{n-2}},\ldots ,x_{n-2}+x_{n}^{N^{2}},x_{n-1}+x_{n}^{N},x_{n})}
  has the form 
  
    
      
        
          x
          
            n
          
          
            m
          
        
      
    
    {\displaystyle x_{n}^{m}}
   for a suitable 
  
    
      
        m
        ∈
        
          N
        
      
    
    {\displaystyle m\in \mathbb {N} }
  .
Proof:
Let 
  
    
      
        
          x
          
            1
          
          
            
              k
              
                1
              
            
          
        
        
          x
          
            2
          
          
            
              k
              
                2
              
            
          
        
        ⋯
        
          x
          
            n
          
          
            
              k
              
                n
              
            
          
        
      
    
    {\displaystyle x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{n}^{k_{n}}}
   be an arbitrary monomial of 
  
    
      
        f
      
    
    {\displaystyle f}
  . Inserting 
  
    
      
        
          x
          
            1
          
        
        +
        
          x
          
            n
          
          
            
              N
              
                n
                −
                1
              
            
          
        
      
    
    {\displaystyle x_{1}+x_{n}^{N^{n-1}}}
   for 
  
    
      
        
          x
          
            1
          
        
      
    
    {\displaystyle x_{1}}
  , 
  
    
      
        
          x
          
            2
          
        
        +
        
          x
          
            n
          
          
            
              N
              
                n
                −
                2
              
            
          
        
      
    
    {\displaystyle x_{2}+x_{n}^{N^{n-2}}}
   for 
  
    
      
        
          x
          
            2
          
        
      
    
    {\displaystyle x_{2}}
   gives

  
    
      
        (
        
          x
          
            1
          
        
        +
        
          x
          
            n
          
          
            
              N
              
                n
                −
                1
              
            
          
        
        
          )
          
            
              k
              
                1
              
            
          
        
        (
        
          x
          
            2
          
        
        +
        
          x
          
            n
          
          
            
              N
              
                n
                −
                2
              
            
          
        
        
          )
          
            
              k
              
                2
              
            
          
        
        ⋯
        (
        
          x
          
            n
            −
            1
          
        
        +
        
          x
          
            n
          
          
            N
          
        
        
          )
          
            
              k
              
                n
                −
                1
              
            
          
        
        
          x
          
            n
          
          
            
              k
              
                n
              
            
          
        
      
    
    {\displaystyle (x_{1}+x_{n}^{N^{n-1}})^{k_{1}}(x_{2}+x_{n}^{N^{n-2}})^{k_{2}}\cdots (x_{n-1}+x_{n}^{N})^{k_{n-1}}x_{n}^{k_{n}}}
  .This is a polynomial, and moreover, by definition 
  
    
      
        g
      
    
    {\displaystyle g}
   consists of certain coefficients multiplied by polynomials of that form.
We want to find the largest coefficient of 
  
    
      
        g
      
    
    {\displaystyle g}
  . To do so, we first identify the largest monomial of

  
    
      
        (
        
          x
          
            1
          
        
        +
        
          x
          
            n
          
          
            
              N
              
                n
                −
                1
              
            
          
        
        
          )
          
            
              k
              
                1
              
            
          
        
        (
        
          x
          
            2
          
        
        +
        
          x
          
            n
          
          
            
              N
              
                n
                −
                2
              
            
          
        
        
          )
          
            
              k
              
                2
              
            
          
        
        ⋯
        (
        
          x
          
            n
            −
            1
          
        
        +
        
          x
          
            n
          
          
            N
          
        
        
          )
          
            
              k
              
                n
                −
                1
              
            
          
        
        
          x
          
            n
          
          
            
              k
              
                n
              
            
          
        
      
    
    {\displaystyle (x_{1}+x_{n}^{N^{n-1}})^{k_{1}}(x_{2}+x_{n}^{N^{n-2}})^{k_{2}}\cdots (x_{n-1}+x_{n}^{N})^{k_{n-1}}x_{n}^{k_{n}}}
  by multiplying out; it turns out, that always choosing 
  
    
      
        
          x
          
            n
          
          
            
              N
              
                j
              
            
          
        
      
    
    {\displaystyle x_{n}^{N^{j}}}
   yields a strictly larger monomial than instead preferring the other variable 
  
    
      
        
          x
          
            j
          
        
      
    
    {\displaystyle x_{j}}
  . Hence, the strictly largest monomial of that polynomial under consideration is

  
    
      
        (
        
          x
          
            n
          
          
            
              N
              
                n
                −
                1
              
            
          
        
        
          )
          
            
              k
              
                1
              
            
          
        
        (
        
          x
          
            n
          
          
            
              N
              
                n
                −
                2
              
            
          
        
        
          )
          
            
              k
              
                2
              
            
          
        
        ⋯
        (
        
          x
          
            n
          
          
            N
          
        
        
          )
          
            
              k
              
                n
                −
                1
              
            
          
        
        
          x
          
            n
          
          
            
              k
              
                n
              
            
          
        
        =
        
          x
          
            n
          
          
            
              k
              
                1
              
            
            
              N
              
                n
                −
                1
              
            
            +
            
              k
              
                2
              
            
            
              N
              
                n
                −
                2
              
            
            +
            ⋯
            +
            
              k
              
                n
                −
                1
              
            
            N
            +
            
              k
              
                n
              
            
          
        
      
    
    {\displaystyle (x_{n}^{N^{n-1}})^{k_{1}}(x_{n}^{N^{n-2}})^{k_{2}}\cdots (x_{n}^{N})^{k_{n-1}}x_{n}^{k_{n}}=x_{n}^{k_{1}N^{n-1}+k_{2}N^{n-2}+\cdots +k_{n-1}N+k_{n}}}
  .Now 
  
    
      
        N
      
    
    {\displaystyle N}
   is larger than all the 
  
    
      
        
          k
          
            j
          
        
      
    
    {\displaystyle k_{j}}
   involved here, since it's even larger than the degree of any monomial of 
  
    
      
        f
      
    
    {\displaystyle f}
  . Therefore, for 
  
    
      
        (
        
          k
          
            1
          
        
        ,
        …
        ,
        
          k
          
            n
          
        
        )
      
    
    {\displaystyle (k_{1},\ldots ,k_{n})}
   coming from monomials of 
  
    
      
        f
      
    
    {\displaystyle f}
  , the numbers

  
    
      
        
          k
          
            1
          
        
        
          N
          
            n
            −
            1
          
        
        +
        
          k
          
            2
          
        
        
          N
          
            n
            −
            2
          
        
        +
        ⋯
        +
        
          k
          
            n
            −
            1
          
        
        N
        +
        
          k
          
            n
          
        
      
    
    {\displaystyle k_{1}N^{n-1}+k_{2}N^{n-2}+\cdots +k_{n-1}N+k_{n}}
  represent numbers in the number system base 
  
    
      
        N
      
    
    {\displaystyle N}
  . In particular, no two of them are equal for distinct 
  
    
      
        (
        
          k
          
            1
          
        
        ,
        …
        ,
        
          k
          
            n
          
        
        )
      
    
    {\displaystyle (k_{1},\ldots ,k_{n})}
  , since numbers of base 
  
    
      
        N
      
    
    {\displaystyle N}
   must have same 
  
    
      
        N
      
    
    {\displaystyle N}
  -cimal places to be equal. Hence, there is a largest of them, call it 
  
    
      
        
          m
          
            1
          
        
        
          N
          
            n
            −
            1
          
        
        +
        
          m
          
            2
          
        
        
          N
          
            n
            −
            2
          
        
        +
        ⋯
        +
        
          m
          
            n
            −
            1
          
        
        N
        +
        
          m
          
            n
          
        
      
    
    {\displaystyle m_{1}N^{n-1}+m_{2}N^{n-2}+\cdots +m_{n-1}N+m_{n}}
  . The largest monomial of

  
    
      
        (
        
          x
          
            1
          
        
        +
        
          x
          
            n
          
          
            
              N
              
                n
                −
                1
              
            
          
        
        
          )
          
            
              m
              
                1
              
            
          
        
        (
        
          x
          
            2
          
        
        +
        
          x
          
            n
          
          
            
              N
              
                n
                −
                2
              
            
          
        
        
          )
          
            
              m
              
                2
              
            
          
        
        ⋯
        (
        
          x
          
            n
            −
            1
          
        
        +
        
          x
          
            n
          
          
            N
          
        
        
          )
          
            
              m
              
                n
                −
                1
              
            
          
        
        
          x
          
            n
          
          
            
              m
              
                n
              
            
          
        
      
    
    {\displaystyle (x_{1}+x_{n}^{N^{n-1}})^{m_{1}}(x_{2}+x_{n}^{N^{n-2}})^{m_{2}}\cdots (x_{n-1}+x_{n}^{N})^{m_{n-1}}x_{n}^{m_{n}}}
  is then

  
    
      
        
          x
          
            n
          
          
            
              m
              
                1
              
            
            
              N
              
                n
                −
                1
              
            
            +
            
              m
              
                2
              
            
            
              N
              
                n
                −
                2
              
            
            +
            ⋯
            +
            
              m
              
                n
                −
                1
              
            
            N
            +
            
              m
              
                n
              
            
          
        
      
    
    {\displaystyle x_{n}^{m_{1}N^{n-1}+m_{2}N^{n-2}+\cdots +m_{n-1}N+m_{n}}}
  ;its size dominates certainly all monomials coming from the monomial of 
  
    
      
        f
      
    
    {\displaystyle f}
   with powers 
  
    
      
        (
        
          m
          
            1
          
        
        ,
        …
        ,
        
          m
          
            n
          
        
        )
      
    
    {\displaystyle (m_{1},\ldots ,m_{n})}
  , and by choice it also dominates the largest monomial of any polynomials generated by any other monomial of 
  
    
      
        f
      
    
    {\displaystyle f}
  . Hence, it is the largest monomial of 
  
    
      
        g
      
    
    {\displaystyle g}
   measured by degree, and it has the desired form.
  
    
      
        ◻
      
    
    {\displaystyle \Box }
  


== Algebraic independence in algebras ==
A notion well-known in the theory of fields extends to algebras.


== Transitivity of localisation ==


== The theorem ==


== Localisation of fields ==