Commutative Algebra/Noether's normalisation lemma

Computational preparation

Lemma 23.1:

Let $R$ be a ring, and let $f\in R[x_1,\dots ,x_n]$ be a polynomial. Let $N\in \mathbb{N}$ be a number that is strictly larger than the degree of any monomial of $f$ (where the degree of an arbitrary monomial $x_1^{k_1}{x}_2^{k_2}\cdots x_n^{k_n}$ of $f$ is defined to be $k_1+k_2+\cdots +k_n$). Then the largest monomial (with respect to degree) of the polynomial

$g(x_1,\dots ,x_n):=f(x_1+x_n^{N^{n-1}},x_2+x_n^{N^{n-2}},\dots ,x_{n-2}+x_n^{N^2},x_{n-1}+x_n^N,x_n)$

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

Proof:

Let $x_1^{k_1}{x}_2^{k_2}\cdots x_n^{k_n}$ be an arbitrary monomial of $f$. Inserting $x_1+x_n^{N^{n-1}}$ for $x_1$, $x_2+x_n^{N^{n-2}}$ for $x_2$ gives

$(x_1+x_n^{N^{n-1}}{)}^{k_1}(x_2+x_n^{N^{n-2}}{)}^{k_2}\cdots (x_{n-1}+x_n^N{)}^{k_{n-1}}{x}_n^{k_n}$.

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

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

$(x_1+x_n^{N^{n-1}}{)}^{k_1}(x_2+x_n^{N^{n-2}}{)}^{k_2}\cdots (x_{n-1}+x_n^N{)}^{k_{n-1}}{x}_n^{k_n}$

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

$(x_n^{N^{n-1}}{)}^{k_1}(x_n^{N^{n-2}}{)}^{k_2}\cdots (x_n^N{)}^{k_{n-1}}{x}_n^{k_n}=x_n^{k_1{N}^{n-1}+k_2{N}^{n-2}+\cdots +k_{n-1}N+k_n}$.

Now $N$ is larger than all the $k_j$ involved here, since it's even larger than the degree of any monomial of $f$. Therefore, for $(k_1,\dots ,k_n)$ coming from monomials of $f$, the numbers

$k_1{N}^{n-1}+k_2{N}^{n-2}+\cdots +k_{n-1}N+k_n$

represent numbers in the number system base $N$. In particular, no two of them are equal for distinct $(k_1,\dots ,k_n)$, since numbers of base $N$ must have same $N$-cimal places to be equal. Hence, there is a largest of them, call it $m_1{N}^{n-1}+m_2{N}^{n-2}+\cdots +m_{n-1}N+m_n$. The largest monomial of

$(x_1+x_n^{N^{n-1}}{)}^{m_1}(x_2+x_n^{N^{n-2}}{)}^{m_2}\cdots (x_{n-1}+x_n^N{)}^{m_{n-1}}{x}_n^{m_n}$

is then

$x_n^{m_1{N}^{n-1}+m_2{N}^{n-2}+\cdots +m_{n-1}N+m_n}$;

its size dominates certainly all monomials coming from the monomial of $f$ with powers $(m_1,\dots ,m_n)$, and by choice it also dominates the largest monomial of any polynomials generated by any other monomial of $f$. Hence, it is the largest monomial of $g$ measured by degree, and it has the desired form.$◻$

Algebraic independence in algebras

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