Noether normalization

Every finitely generated algebra $A$ over a field has algebraically independent elements $y_1,\dots ,y_d$ such that $A$ is finite as a module over the polynomial subalgebra $k[y_1,\dots ,y_d]$. The integer $d$ is the Krull dimension of $A$.

For a finite integral inclusion $A=k[t_1,\dots ,t_r]\subset B$ of domains, a nonzero $b\in B$ has a monic minimal polynomial over the fraction field with nonzero constant term $a_0\in A$. Its equation gives $a_0\in bB\cap A$. Consequently the image of $V_B(b)$ under the finite morphism to ${\mathbb{A}}^r$ is contained in the proper Zariski-closed set $V_A(a_0)$. Finite morphisms preserve the dimension of a closed subvariety and its image, so this proves dimension drops for proper closed subsets after Noether normalization. A power of $a_0$, up to sign, is the field norm of $b$.

For a nonconstant polynomial $f\in k[t_1,\dots ,t_n]$ over an infinite field, an invertible linear change of coordinates can make $f$ monic in the final coordinate after multiplication by a scalar. Therefore $k[t_1,\dots ,t_n]/(f)$ is integral over a polynomial algebra in the first $n-1$ new coordinates.