TITLE: Maximal Ideals in the ring k[x1,...,xn ]
QUESTION [20 upvotes]: Hi. From one of the forms of Hilbert's Nullstellensatz we know that all the maximal ideals in a polynomial ring $k[x_1, \dots, x_n]$ where $k$ is an algebraically closed field, are of the form $(x_1 - a_1, \dots , x_n - a_n)$. So that any maximal ideal in this case is generated by polynomials $g_j \in k[x_1, \dots, x_n]$ for $j = 1, \dots , n$, where $g_j$ only depends on the variable $x_j$ (Obviously by taking $g_j = x_j - a_j$). Now, I'm interested in the case of a polynomial ring $k[x_1, \dots, x_n]$ where $k$ is an arbitrary field (i.e., I can't make use of the Nullstellensatz). I suppose this may no longer be the case, i.e., I don't expect any maximal ideal to be generated by n polynomials, each of them only dependent on one of the variables $x_j$, but my question is if maybe the maximal ideals can be generated by polynomials $g_j$ that only depend on the first $j$ variables $x_1, \dots , x_j$? If so, does anybody know how to prove this or can anyone suggest me some references that may help me?. Thanks.

REPLY [6 votes]: In response to the OP's request for an explicit reference:
see Theorems 128 and 130 on pages 82-83 of
http://math.uga.edu/~pete/integral.pdf