TITLE: Polynomial algebra and its special ideals
QUESTION [8 upvotes]: Consider a polynomial algebra $A=\mathbb{K}[x_1,\ldots,x_n]$ and its ideal $I$, such that $A/I=\mathbb{K}[y_1,\ldots, y_k]$. Is it true that there exist new polynomial generators $z_1,\ldots,z_n$ (in a sense that $A=\mathbb{K}[z_1,\ldots,z_n]$) such that $I=(z_1,\ldots,z_{n-k})$?

REPLY [9 votes]: This is called the Embedding Problem. I believe that it is false in positive characteristics and unknown in characteristic zero, except a few small cases. See Kraft's review
for the extent of my knowledge of the state of the problem. The counterexample in characteristic $p$ is from this paper:
$$
K[x,y]/(y^{p^2}-x-x^{2p}) 
$$