Help to understand Noether Normalization Theorem

Question

I'm trying to understand the statement of the Noether Normalization Theorem:

How can $k[x_1,\ldots,x_n]$ be equal to $k[x]$? a typical element of $k[x]$ is for example $ax^2+bx$ with $a,b\in k$, while an element of $k[x_1,\ldots,x_n]$ is for example $cx_1^3x_2x_3^2+x_1^2+d$ with $c,d\in k$.

Thanks in advance.

Answer 1

Here $x$ is just a (relatively common) shorthand for the entire list of variables $x_1, \ldots, x_n$. Typically this is denoted $x_1, \ldots, x_n = \underline{x}$, but for whatever reason the author chose not to denote it this way.