Maximal ideals in rings of polynomials Let $k$ be a field and $D = k[X_1, . . . , X_n]$ the polynomial ring in $n$ variables over $k$.
Show that:
a) Every maximal ideal of $D$ is generated by $n$ elements.
b) If $R$ is ring and $\mathfrak m\subset D=R[X_1,\dots,X_n]$ is maximal ideal such that $\mathfrak m \cap R$ is maximal and generated by $s$ elements, then $\mathfrak m$ is generated by $s + n$ elements.
The days that I am trying to solve. Help me.

We induct on $n$, the result $n=1$ being clear since $k[X_1]$ is a PID. Now consider the inclusion
$$k[X_1,\ldots,X_{n-1}] \subseteq K[X_1,\ldots,X_{n-1}][X_n].$$
Choose a maximal ideal $\mathfrak{m}$ in the ring on the right. Then by Zariski's lemma we have that $k[X_1,\ldots,X_n]/\mathfrak{m}$ is a finite algebraic extension of $k$. Since we have the inclusion
$$k \hookrightarrow k[X_1,\ldots,X_{n-1}]/\mathfrak{m}^c \hookrightarrow k[X_1,\ldots,X_n]/\mathfrak{m}$$
it follows that $k[X_1,\ldots,X_{n-1}]/\mathfrak{m}^c$ is an integral domain that is also a finite dimensional vector space, thus a field. It will now follow that $\mathfrak{m}^c$ is maximal and the induction hypothesis implies that it is generated by $n-1$ elements.  We now need a Lemma (as suggested by Jeff Tolliver below):


Lemma: $$\left(k[X_1,\ldots,X_{n-1}]/\mathfrak{m}^c\right) [X_n] \cong k[X_1,\ldots,X_n]/\mathfrak{m}^ck[X_1,\ldots,X_n]$$
Proof: $$\begin{eqnarray*} \left(k[X_1,\ldots,X_{n-1}]/\mathfrak{m}^c\right) [X_n] &\cong& k[X_1,\ldots,X_{n-1}][X_n] \otimes_{k[X_1,\ldots,X_{n-1}]}  k[X_1,\ldots,X_{n-1}]/\mathfrak{m}^c \\
&\cong& k[X_1,\ldots,X_n]/\mathfrak{m}^c k[X_1,\ldots,X_n] \hspace{5mm} \end{eqnarray*}$$
    where the first line is using exercise 3.6 of Atiyah - Macdonald, the second line exercise 3.2 of the same book. Explicitly if we trace through the isomorphisms, this sends
    $$\overline{a_0} + \overline{a_1}X_n + \ldots + \overline{a_k}X_n^k \mapsto \overline{a_0 + a_1X_n\ldots + a_kX_n^k}$$
    where the bar on the left is the residue class mod $\mathfrak{m}^c$ and on the right mod $\mathfrak{m}^ck[X_1,\ldots,X_n]$. Thus the lemma is proven.


Back to the problem. It is clear that $\mathfrak{m}$ corresponds to a maximal ideal $\overline{\mathfrak{m}}$ in $k[X_1,\ldots,X_n]/\mathfrak{m}^ck[X_1,\ldots,X_n]$ and by the lemma to some maximal ideal in $\left(k[X_1,\ldots,X_{n-1}]/\mathfrak{m}^c\right)[X_n]$. Since this is the polynomial ring over a field, this maximal ideal is generated by some $P = \overline{a_0} + \ldots + \overline{a_k}X_n^k$. Then in $k[X_1,\ldots,X_n]/\mathfrak{m}^ck[X_1,\ldots,X_n]$, the element corresponding to $P$ is $\overline{a_0 + \ldots + a_kX_n^k}$ and this we know generates $\overline{\mathfrak{m}}$. If we write $$f = a_0 + \ldots + a_kX_n^k$$ it will now follow by the induction hypothesis that $\mathfrak{m}$ is generated by $f$ and the $n-1$ elements of $\mathfrak{m}^c$. This completes the proof of the problem.
