The elements of a regular sequence in $k[x\_1, \ldots, x\_n]$ are algebraically independent over $k$ (see for example Matsumura ex. 16.6), and so for a length n regular sequence $(f\_i)$ of homogeneous elements, $k(x\_1, \ldots, x\_n)$ will be algebraic over $k(f\_1, \ldots, f\_n)$.


My question is whether it's also true that $k[x\_1, \ldots, x\_n]$ is integral over $k[f\_1, \ldots, f\_n]$. Aside from experimental evidence (dozens of randomly generated regular sequences in Macaulay with n=3 and low-ish degrees), I don't really have any reason to think that it should be true, and I haven't been able to prove or disprove it (or find a mention of it anywhere).


Thanks.


Edit: added homogeneous condition



Is $k[x_1, \ldots, x_n]$ always an integral extension of $k[f_1, \ldots, f_n]$ for a regular sequence $(f_1, \ldots, f_n)$?
No. For example, the element $y \in k[x,y]$ is not integral over $k[x-1, xy]$. (To see this, write $k[x-1,xy] = k[x,xy]$ and reduce modulo $x$).


