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Lemma 10.2

[lemma] Let $f \in K[x_1, \ldots, x_n]$ be a nonzero polynomial over an infinite field $K$. Assume that $f$ is homomgeneous. Then there are $a_1, \ldots, a_{n-1}$ such that $f(a_1, \ldots, a_{n-1}, 1) \neq 0$. [/lemma]

[example=Example] Consider $f = xy^2 + x^2z \in \RR[x, y, z]$. Then $f(1, 1, 1)$ works. [/example]

[example=Counterexample over finite field] Over $K = \ZZ_2$, this is false. Consider $f = x^2 + xy$. Then $f(x, 1) = x^2 + x$ which is the classic counterexample of a polynomial which vanishes on its entire coefficient ring. [/example]

[proof=Proof of Lemma 10.2] We proceed by induction. The base case $n = 1$ is trivial. So now assume $n > 1$ and write

where the $f_i \in K[x_2, \ldots, x_{n}]$ are homogeneous of degree $d - i$. Since $f$ is nonzero, at least one $f_i$ is nonzero and the inductive hypothesis implies that we can choose $a_2, \ldots, a_{n-1}$ so that

for this $i$. Thus,

is a nonzero polynomial so it has finitely many roots. Since $K$ is infinite, we must be able to find some $a_1$ so that $f(a_1, \ldots, a_{n-1}, 1)\neq 0$. [/proof]

Lemma 10.3

[lemma] Let $f \in K[x_1, \ldots, x_n]$ be a nonzero polynomial over an infinite field $K$. Then there are $\lambda \in K$ and $a_1, \ldots, a_{n-1} \in K$ such that

is monic in $y_n$. [/lemma]

Proposition 10.5 (Noether normalization)

[proposition] Let $R$ be a finitely-generated algebra over a field $K$ with generators $x_1, \ldots, x_n \in R$. Then there is an injective $K$-algebra homomorphism $K[z_1, \ldots, z_r] \to R$ from a polynomial ring over $K$ to $R$ that makes $R$ into a finite extension ring of $K[z_1, \ldots, z_r]$.

Moreover, if $K$ is an infinite field, the images of $z_1, \ldots, z_r$ in $R$ can be chosen to be $K$-linear combinations of $x_1, \ldots, x_n$. [/proposition]