# Commutative Algebra/Noether's normalisation lemma

## Computational preparation

Lemma 23.1:

Let ${\displaystyle R}$ be a ring, and let ${\displaystyle f\in R[x_{1},\ldots ,x_{n}]}$ be a polynomial. Let ${\displaystyle N\in \mathbb {N} }$ be a number that is strictly larger than the degree of any monomial of ${\displaystyle f}$ (where the degree of an arbitrary monomial ${\displaystyle x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{n}^{k_{n}}}$ of ${\displaystyle f}$ is defined to be ${\displaystyle k_{1}+k_{2}+\cdots +k_{n}}$). Then the largest monomial (with respect to degree) of the polynomial

${\displaystyle g(x_{1},\ldots ,x_{n}):=f(x_{1}+x_{n}^{N^{n-1}},x_{2}+x_{n}^{N^{n-2}},\ldots ,x_{n-2}+x_{n}^{N^{2}},x_{n-1}+x_{n}^{N},x_{n})}$

has the form ${\displaystyle x_{n}^{m}}$ for a suitable ${\displaystyle m\in \mathbb {N} }$.

Proof:

Let ${\displaystyle x_{1}^{k_{1}}x_{2}^{k_{2}}\cdots x_{n}^{k_{n}}}$ be an arbitrary monomial of ${\displaystyle f}$. Inserting ${\displaystyle x_{1}+x_{n}^{N^{n-1}}}$ for ${\displaystyle x_{1}}$, ${\displaystyle x_{2}+x_{n}^{N^{n-2}}}$ for ${\displaystyle x_{2}}$ gives

${\displaystyle (x_{1}+x_{n}^{N^{n-1}})^{k_{1}}(x_{2}+x_{n}^{N^{n-2}})^{k_{2}}\cdots (x_{n-1}+x_{n}^{N})^{k_{n-1}}x_{n}^{k_{n}}}$.

This is a polynomial, and moreover, by definition ${\displaystyle g}$ consists of certain coefficients multiplied by polynomials of that form.

We want to find the largest coefficient of ${\displaystyle g}$. To do so, we first identify the largest monomial of

${\displaystyle (x_{1}+x_{n}^{N^{n-1}})^{k_{1}}(x_{2}+x_{n}^{N^{n-2}})^{k_{2}}\cdots (x_{n-1}+x_{n}^{N})^{k_{n-1}}x_{n}^{k_{n}}}$

by multiplying out; it turns out, that always choosing ${\displaystyle x_{n}^{N^{j}}}$ yields a strictly larger monomial than instead preferring the other variable ${\displaystyle x_{j}}$. Hence, the strictly largest monomial of that polynomial under consideration is

${\displaystyle (x_{n}^{N^{n-1}})^{k_{1}}(x_{n}^{N^{n-2}})^{k_{2}}\cdots (x_{n}^{N})^{k_{n-1}}x_{n}^{k_{n}}=x_{n}^{k_{1}N^{n-1}+k_{2}N^{n-2}+\cdots +k_{n-1}N+k_{n}}}$.

Now ${\displaystyle N}$ is larger than all the ${\displaystyle k_{j}}$ involved here, since it's even larger than the degree of any monomial of ${\displaystyle f}$. Therefore, for ${\displaystyle (k_{1},\ldots ,k_{n})}$ coming from monomials of ${\displaystyle f}$, the numbers

${\displaystyle k_{1}N^{n-1}+k_{2}N^{n-2}+\cdots +k_{n-1}N+k_{n}}$

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

${\displaystyle (x_{1}+x_{n}^{N^{n-1}})^{m_{1}}(x_{2}+x_{n}^{N^{n-2}})^{m_{2}}\cdots (x_{n-1}+x_{n}^{N})^{m_{n-1}}x_{n}^{m_{n}}}$

is then

${\displaystyle x_{n}^{m_{1}N^{n-1}+m_{2}N^{n-2}+\cdots +m_{n-1}N+m_{n}}}$;

its size dominates certainly all monomials coming from the monomial of ${\displaystyle f}$ with powers ${\displaystyle (m_{1},\ldots ,m_{n})}$, and by choice it also dominates the largest monomial of any polynomials generated by any other monomial of ${\displaystyle f}$. Hence, it is the largest monomial of ${\displaystyle g}$ 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.

Theorem 23.2:

Let ${\displaystyle R}$ be a ring and ${\displaystyle A}$ an ${\displaystyle R}$-algebra. Elements ${\displaystyle a_{1},\ldots ,a_{n}}$ in ${\displaystyle A}$ are called algebraically independent over ${\displaystyle R}$ iff there does not exist a polynomial ${\displaystyle p\in R[x_{1},\ldots ,x_{n}]}$ such that ${\displaystyle p(a_{1},\ldots ,a_{n})=0}$ (where the polynomial is evaluated as explained in chapter 21).

## The theorem

Theorem 23.3 (Noether's normalisation lemma):

Let ${\displaystyle D}$ be an integral domain, and let ${\displaystyle E\supseteq D}$ be a ring extension of ${\displaystyle D}$ that is finitely generated as a ${\displaystyle D}$-module; in particular, ${\displaystyle E}$ is a ${\displaystyle D}$-algebra, where the algebra operations are induced by the ring operations. Then we may pick a ${\displaystyle d\in D}$ such that there exist ${\displaystyle c_{1},\ldots ,c_{n}\in E_{d}}$ (${\displaystyle E_{d}}$ denoting the localisation of ${\displaystyle E_{d}}$ at ${\displaystyle d}$) which are algebraically independent over ${\displaystyle E_{d}}$ as a ${\displaystyle D_{d}}$-algebra