[<< 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 ==