Polynomial rings are regular I want to use the result that the polynomial ring $R:=k[x_1,\cdots,x_n]$ is a regular ring.
I can prove that every maximal ideal $\mathfrak m$ in $R$ can be generated by $n$ number of elements so that $R_{\mathfrak m}$ is a regular local ring.
Now Serre's result shows that if $A$ is a regular local ring then $A_{\mathfrak p}$ is a regular local ring. From this it follows that $R$ is a regular ring. Now Serre's result is a BIG theorem, so is there any other easier way to prove this result. What if we assume $k$ is algebraically closed?
I have a presentation $+$ viva (on some topic in commutative algebra) coming up next week and I need to use the above result. So what I kind of question I can expect when I use this theorem?
I know I have asked many questions in one post but these are all related. I am sorry for this.
Thank you.
 A: This can be proved if you know Noether Normalization, which essentially uses the fact any non-constant polynomial in $n$ variables can be made monic in one of the variables after a change of variables. So, let us assume this.
Then, maximal ideals are $n$ generated follows by induction on $n$ as follows. $n=1$ being trivial, assume proved for $n-1$ and let $M\subset R=k[x_1,\ldots, x_n]$ be a maximal ideal. Then, after change of variables, we may assume (since $M\neq 0$), $M$ contains a monic polynomial in $x_n$. Let $N=M\cap S$ where $S=k[x_1,\ldots, x_{n-1}]$. We have thus an integral extension $S/N\to R/M$, but $R/M$ is a field implies by integrality, that so is $S/N$. So, $N$ is a maximal ideal of $S$ and so by induction generated by $n-1$ elements. Then, the image of $M$ in $S/N[x_n]$ is a maximal ideal and so by one variable case, we get that it is one generated, and thus $M$ itself is $n$-generated. 
For a prime ideal $P$, we can by Noether Normalization, assume that $k[x_1,\ldots,x_k]\subset R/P$ is an integral extension where height of $P=n-k$. This says, in particular, $P\cap k[x_1,\ldots,x_k]=0$ and thus $R_P$ is a localisation of $k(x_1,\ldots,x_k)[x_{k+1},\ldots,x_n]$ at the image of $P$. The integrality says the image of $P$ in this ring is a maximal ideal and thus, by the previous part, generated by $n-k$ elements which  implies, so is the localization. Hope rest is clear.
