Commutative Algebra/Noether's normalisation lemma

Computational preparation
Lemma 23.1:

Let $$R$$ be a ring, and let $$f \in R[x_1, \ldots, x_n]$$ 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
 * $$g(x_1, \ldots, x_n) := f(x_1 + x_n^{N^{n-1}}, x_2 + x_n^{N^{n-2}}, \ldots, x_{n-2} + x_n^{N^2}, x_{n-1} + x_n^N, x_n)$$

has the form $$x_n^m$$ 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 $$(k_1, \ldots, k_n)$$ 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 $$(k_1, \ldots, k_n)$$, 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 $$(m_1, \ldots, m_n)$$, 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.